A Math

A Math (IGCSE)

E Math

H1 Math

H2 Math

IB HL Math

IB SL Math

IP Math

<p>A rice distributor supplies rice to $4$ different supermarkets in Singapore. The number of bags of rice supplied per delivery to each supermarket, the sizes and cost price are shown in the table below.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 400px;"><colgroup><col style="width: 120px;"><col style="width: 60px;"><col style="width: 60px;"><col style="width: 60px;"><col style="width: 100px;"></colgroup><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></th><th colspan="3" style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Number of bags of rice per delivery</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Number of deliveries in one month</th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start">Size of each bag</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$2.5$ kg</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$5$ kg</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$10$ kg</td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start">Supermarket $A$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$40$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$25$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$10$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$6$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start">Supermarket $B$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$80$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$60$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$30$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$8$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start">Supermarket $C$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$30$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$20$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$25$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$11$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start">Supermarket $D$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$40$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$35$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$30$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$9$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start">Price per bag</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$\$4$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$\$7$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$\$12$</p></td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td></tr></tbody></table></div></div>
<ol class="abc-list"><li><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">If&nbsp;$\mathbf{P} = \begin{pmatrix} 40 & 25 & 10 \end{pmatrix}$, evaluate $6\mathbf{P}$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Describe what is represented by the elements of $6\mathbf{P}$.</span></div></li></ol></li><li><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">If&nbsp;$\mathbf{Q} = \begin{pmatrix} 40 & 25 & 10 \\ 80 & 60 & 30 \\ 30 & 20 & 25 \\ 40 & 35 & 30 \end{pmatrix}$ and $\mathbf{R} = \begin{pmatrix} 4 \\ 7 \\ 12 \end{pmatrix}$, evaluate $\mathbf{QR}$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Describe what is represented by the elements of $\mathbf{QR}$.</span></div></li></ol></li><li><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down two matrices such that the product under the matrix multiplication give the total number of each size of bag the distributor supply to the four supermarkets in one month.</span></div></li><li><div class="list-item-paragraph">Evaluate the product of the two matrices in <strong>(c)(i)</strong>.</div></li></ol></li></ol>
<p>Using the product from part <strong>(c)(ii)</strong> and multiplying by another matrix, evaluate the total amount of money the distributor received from the four supermarkets in one month.</p>

A rice distributor supplies rice to $4$ different supermarkets in Singapore. The number of bags of rice supplied per delivery to each supermarket, the sizes and cost price are shown in the table below.

Using the product from part **(c)(ii)** and multiplying by another matrix, evaluate the total amount of money the distributor received from the four supermarkets in one month.

Application of Matrices

Matrices

<p>Ashlyn, Hayden and Kahlan took a multiple choice test which consists of $40$ questions.</p>
<p>Kahlan attempted all the questions.</p>
<p>The table shows the results of the test and the marks awarded.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: left; --cell-justify: flex-start"></th><th style="text-align: center; --cell-justify: center">Correct</th><th style="text-align: center; --cell-justify: center">Incorrect</th><th style="text-align: center; --cell-justify: center">Not attempted</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">Ashlyn</td><td style="text-align: center; --cell-justify: center">$30$</td><td style="text-align: center; --cell-justify: center">$6$</td><td style="text-align: center; --cell-justify: center">$4$</td></tr><tr><td style="text-align: center; --cell-justify: center">Hayden</td><td style="text-align: center; --cell-justify: center">$32$</td><td style="text-align: center; --cell-justify: center">$7$</td><td style="text-align: center; --cell-justify: center">$1$</td></tr><tr><td style="text-align: center; --cell-justify: center">Kahlan</td><td style="text-align: center; --cell-justify: center">$x$</td><td style="text-align: center; --cell-justify: center">$40 - x$</td><td style="text-align: center; --cell-justify: center">$0$</td></tr><tr><td style="text-align: center; --cell-justify: center">Marks</td><td style="text-align: center; --cell-justify: center">$2$</td><td style="text-align: center; --cell-justify: center">$-1$</td><td style="text-align: center; --cell-justify: center">$0$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-content"><span class="list-item-text">Evaluate $\begin{pmatrix} 30 & 6 & 4 \\ 32 & 7 & 1 \\ x & 40 - x & 0 \end{pmatrix} \begin{pmatrix} 2 \\ -1 \\ 0 \end{pmatrix}$.</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-content"><span class="list-item-text">Explain what your answer to <strong>(a)</strong> represents.</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-content"><span class="list-item-text">If Kahlan has scored the highest marks, using your answer in part <strong>(a)</strong>, find the least number of correct answers she has scored.</span></div></li></ol>

Ashlyn, Hayden and Kahlan took a multiple choice test which consists of $40$ questions.

The table shows the results of the test and the marks awarded.

Evaluate $\begin{pmatrix} 30 & 6 & 4 \\ 32 & 7 & 1 \\ x & 40 - x & 0 \end{pmatrix} \begin{pmatrix} 2 \\ -1 \\ 0 \end{pmatrix}$.

Explain what your answer to **(a)** represents.

If Kahlan has scored the highest marks, using your answer in part **(a)**, find the least number of correct answers she has scored.

<p>It is given that&nbsp;$\mathbf{M} = \begin{pmatrix} 2p & q \\ 0 & \frac{1}{3}r \end{pmatrix}$ and $\mathbf{N} = \begin{pmatrix} 4 & -2 \\ 0 & 6 \end{pmatrix}$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find $\mathbf{N}^2$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$4\mathbf{M} = \mathbf{N}^2$, find the values of $p$, $q$ and $r$.</span></div></li></ol>

It is given that $\mathbf{M} = \begin{pmatrix} 2p & q \\ 0 & \frac{1}{3}r \end{pmatrix}$ and $\mathbf{N} = \begin{pmatrix} 4 & -2 \\ 0 & 6 \end{pmatrix}$.

Given that $4\mathbf{M} = \mathbf{N}^2$, find the values of $p$, $q$ and $r$.

<p>Given that&nbsp;$\mathbf{A} = \begin{pmatrix} 3 & -1 \\ -2 & 1 \end{pmatrix}$, $\mathbf{B} = \begin{pmatrix} 2 & k \\ 4 & 1 \end{pmatrix}$ and $\mathbf{C} = \begin{pmatrix} 3 & 4 \\ 0 & 1 \end{pmatrix}$,</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">find $\mathbf{A}^2 + 2\mathbf{B}$ in terms of $k$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">If&nbsp;$\mathbf{A}^2 + 2\mathbf{B} = 5\mathbf{C}$, find the value of $k$.</span></div></li></ol>

Given that $\mathbf{A} = \begin{pmatrix} 3 & -1 \\ -2 & 1 \end{pmatrix}$, $\mathbf{B} = \begin{pmatrix} 2 & k \\ 4 & 1 \end{pmatrix}$ and $\mathbf{C} = \begin{pmatrix} 3 & 4 \\ 0 & 1 \end{pmatrix}$,

find $\mathbf{A}^2 + 2\mathbf{B}$ in terms of $k$.

If $\mathbf{A}^2 + 2\mathbf{B} = 5\mathbf{C}$, find the value of $k$.

<p>It is given that matrices $\mathbf{P} = \begin{pmatrix} 3y & 1 \\ 2 & 3 \end{pmatrix}$, $\mathbf{Q} = \begin{pmatrix} y & 4 \\ 8 & 2 \end{pmatrix}$ and $\mathbf{R} = \begin{pmatrix} 5 & 1 \\ 2 & 8 \end{pmatrix}$.</p>
<p>If&nbsp;$3\mathbf{P} - \frac{1}{2}\mathbf{Q} = \mathbf{R}$, find the value of $y$.</p>

It is given that matrices $\mathbf{P} = \begin{pmatrix} 3y & 1 \\ 2 & 3 \end{pmatrix}$, $\mathbf{Q} = \begin{pmatrix} y & 4 \\ 8 & 2 \end{pmatrix}$ and $\mathbf{R} = \begin{pmatrix} 5 & 1 \\ 2 & 8 \end{pmatrix}$.

If $3\mathbf{P} - \frac{1}{2}\mathbf{Q} = \mathbf{R}$, find the value of $y$.

<p>Given that&nbsp;$\begin{pmatrix} 3 & h+2 \\ 3 & k \end{pmatrix} + \begin{pmatrix} 2 & -5 \\ 3 & 3k+1 \end{pmatrix} = \begin{pmatrix} 5 & 17-h \\ 6 & 3 \end{pmatrix}$, find the values of $h$ and $k$.</p>

Given that $\begin{pmatrix} 3 & h+2 \\ 3 & k \end{pmatrix} + \begin{pmatrix} 2 & -5 \\ 3 & 3k+1 \end{pmatrix} = \begin{pmatrix} 5 & 17-h \\ 6 & 3 \end{pmatrix}$, find the values of $h$ and $k$.

<p>For any square matrix $\mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, where&nbsp;$a$, $b$, $c$ and $d$ are real numbers, the <strong>determinant</strong> of $\mathbf{A}$ is denoted by</p>
<p style="text-align: center">$|\mathbf{A}| = ad - bc .$</p>
<p>If&nbsp;$|\mathbf{A}| \neq 0$, the <strong>inverse matrix</strong> of $\mathbf{A}$ denoted by</p>
<p style="text-align: center">\[\mathbf{A}^{-1} = \frac{1}{|\mathbf{A}|} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix},\]</p>
<p>such that&nbsp;$\mathbf{A}\mathbf{A}^{-1} = \mathbf{A}^{-1}\mathbf{A} = \mathbf{I}$, where&nbsp;$\mathbf{I}$ is the identity matrix.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the inverse of: $\mathbf{A} = \begin{pmatrix} 2 & 1 \\ 5 & -3 \end{pmatrix}$.</div></li><li><div class="list-item-paragraph">Hence, solve the system of equations<br></div><div class="list-item-paragraph" style="text-align: center">$ \;\,\, 2x + y = 5$<br>$5x - 3y = 7$</div></li></ol>

For any square matrix $\mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, where $a$, $b$, $c$ and $d$ are real numbers, the **determinant** of $\mathbf{A}$ is denoted by

If $|\mathbf{A}| \neq 0$, the **inverse matrix** of $\mathbf{A}$ denoted by

\[\mathbf{A}^{-1} = \frac{1}{|\mathbf{A}|} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix},\]

such that $\mathbf{A}\mathbf{A}^{-1} = \mathbf{A}^{-1}\mathbf{A} = \mathbf{I}$, where $\mathbf{I}$ is the identity matrix.

Find the inverse of: $\mathbf{A} = \begin{pmatrix} 2 & 1 \ 5 & -3 \end{pmatrix}$.

Hence, solve the system of equations\n\[\begin{aligned} 2x + y &= 5 \ 5x - 3y &= 7 \end{aligned}\]

IP3U8 HOT Q1

<p>The table below shows the number of tubs of ice cream of four different brands: $A$, $B$, $C$ and $D$, sold at $3$ dessert shops.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 500px;"><colgroup><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"></colgroup><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"> </th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Brand $A$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Brand $B$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Brand $C$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Brand $D$</th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Shop $1$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$5$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$3$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$1$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$1$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Shop $2$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$11$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$6$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$8$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$5$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Shop $3$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$8$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$5$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$4$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$3$</td></tr></tbody></table></div></div>
<p>A tub of ice cream costs $\$15$ for Brand $A$, $\$10$ for Brand $B$, $\$8$ for Brand $C$ and $\$5$ for Brand $D$.</p>
<p>Given the matrix $P = \begin{pmatrix} 15 \\ 10 \\ 8 \\ 5 \end{pmatrix}$,</p>
<ol class="abc-list"><li><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down a $3 \times 4$ matrix $Q$ that represents the data in the table above.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Evaluate $QP$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Explain what the elements in $QP$ represent.</span></div></li></ol></li></ol>
<p>Brand $A$ comes in tubs of $180$ g, Brand $B$ in tubs of $200$ g, Brand $C$ in tubs of $150$ g and Brand $D$ in tubs of $100$ g.</p>
<ol class="abc-list" start="2"><li><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down a matrix $S$, such that matrix $QS$ represents the total number of grams of ice cream sold by each shop.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Evaluate $QS$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Using matrix multiplication, find the total number of grams of ice cream sold by the $3$ shops.</span></div></li></ol></li></ol>

The table below shows the number of tubs of ice cream of four different brands: $A$, $B$, $C$ and $D$, sold at $3$ dessert shops.

A tub of ice cream costs $\$15$ for Brand $A$, $\$10$ for Brand $B$, $\$8$ for Brand $C$ and $\$5$ for Brand $D$.

Given the matrix $P = \begin{pmatrix} 15 \\ 10 \\ 8 \\ 5 \end{pmatrix}$,

Brand $A$ comes in tubs of $180$ g, Brand $B$ in tubs of $200$ g, Brand $C$ in tubs of $150$ g and Brand $D$ in tubs of $100$ g.

Mathematical Modeling for Inventory Management

<p>The table below records the number of $\$2$, $\$5$ and $\$10$ notes $3$ children, Alvin, Brian and Colin has.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 400px;"><colgroup><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"></colgroup><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"> </th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$\$2$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$\$5$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$\$10$</th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Alvin</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$3$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$1$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$1$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Brian</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$1$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$3$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$0$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Colin</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$0$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$2$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$3$</td></tr></tbody></table></div></div>
<p>The information in the table can be represented by a matrix $P$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Write down the matrix $P$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Calculate $PS$ where&nbsp;$S = \begin{pmatrix} 2 \\ 5 \\ 10 \end{pmatrix}$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Describe what is represented by the elements of $PS$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$Q = \begin{pmatrix} 1 & 1 & 1 \end{pmatrix}$, calculate $R = QP$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Describe what is represented by the elements of $R$.</span></div></li></ol>

The table below records the number of $\$2$, $\$5$ and $\$10$ notes $3$ children, Alvin, Brian and Colin has.

The information in the table can be represented by a matrix $P$.

Calculate $PS$ where $S = \begin{pmatrix} 2 \\ 5 \\ 10 \end{pmatrix}$.

Describe what is represented by the elements of $PS$.

Given that $Q = \begin{pmatrix} 1 & 1 & 1 \end{pmatrix}$, calculate $R = QP$.

Describe what is represented by the elements of $R$.

Mathematical Modeling with Matrices

<p>Given that&nbsp;$A = \begin{pmatrix} -2 & 1 \\ -1 & 4 \end{pmatrix}$, $B = \begin{pmatrix} 3 & 1 \\ -2 & -5 \end{pmatrix}$ and $C = \begin{pmatrix} 3 & -1 \\ 4 & -2 \end{pmatrix}$, write $AB + C^2$ as a single matrix.</p>

Given that $A = \begin{pmatrix} -2 & 1 \\ -1 & 4 \end{pmatrix}$, $B = \begin{pmatrix} 3 & 1 \\ -2 & -5 \end{pmatrix}$ and $C = \begin{pmatrix} 3 & -1 \\ 4 & -2 \end{pmatrix}$, write $AB + C^2$ as a single matrix.

Multiplication of Two Matrices

Matrix Arithmetic: Computing the Square of a Matrix

<p>Given that $A = \begin{pmatrix} 2 & 0 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} x & 0 \\ 2 & 1 \end{pmatrix}$, find the value of $x$ when $AB = BA$.</p>

Given that $A = \begin{pmatrix} 2 & 0 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} x & 0 \\ 2 & 1 \end{pmatrix}$, find the value of $x$ when $AB = BA$.

Properties of Matrix Multiplication: Commutativity

<p>Given that&nbsp;$A = \begin{pmatrix} 1 & 0 & 1 \end{pmatrix}$, $B = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$ and $C = \begin{pmatrix} 1 & -2 \\ 2 & 0 \end{pmatrix}$, find</p>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-content"><span class="list-item-text">$AB$,</span></div></li><li data-list-item-id="item-b"><div class="list-item-content"><span class="list-item-text">$C^2$.</span></div></li></ol>

Given that $A = \begin{pmatrix} 1 & 0 & 1 \end{pmatrix}$, $B = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$ and $C = \begin{pmatrix} 1 & -2 \\ 2 & 0 \end{pmatrix}$, find

Operations on Matrices: Products and Powers

<div class="content-block content-block--table"><div class="table-container"><table><colgroup><col style="width: 650px;"></colgroup><tbody><tr style="height: 43px;"><td rowspan="2" style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"><p><strong>Tips – Multiplication of Two Matrices</strong></p><ul><li><p>Two matrices $A$ and $B$ could only be multiplied, $A \times B$ or $AB$, </p></li></ul><p>$\quad\,$ if numbers of columns of $A =$ number of rows of $B$.</p><p>$\quad\,$ If $A$ is a $m \times n$ matrix and $B$ is an $n \times b$, the result product will be a $m \times b$ matrix.</p><p>$\quad\,$ Example: $\begin{pmatrix} 2 &amp; 3 \end{pmatrix} \begin{pmatrix} 1 &amp; 2 \\ 3 &amp; 4 \end{pmatrix} = \begin{pmatrix} 11 &amp; 16 \end{pmatrix}$</p><ul><li><p>Multiplication of matrices is not commutative $\rightarrow AB \neq BA$.</p></li></ul></td></tr><tr style="height: 40px;"></tr></tbody></table></div></div>
<p>Express each of the following as a single matrix.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">$\begin{pmatrix}7 & 9\end{pmatrix}\begin{pmatrix}1 & -2\\ 6 & 2\end{pmatrix}$</div></li><li><div class="list-item-paragraph">$\begin{pmatrix}1 & -2\\ 6 & 2\end{pmatrix}\begin{pmatrix}7\\ 9\end{pmatrix}$</div></li></ol>

<p><strong>Tips – Multiplication of Two Matrices</strong></p><ul><li><p>Two matrices $A$ and $B$ could only be multiplied, $A \times B$ or $AB$, </p></li></ul><p>$\quad\,$ if numbers of columns of $A =$ number of rows of $B$.</p><p>$\quad\,$ If $A$ is a $m \times n$ matrix and $B$ is an $n \times b$, the result product will be a $m \times b$ matrix.</p><p>$\quad\,$ Example: $\begin{pmatrix} 2 &amp; 3 \end{pmatrix} \begin{pmatrix} 1 &amp; 2 \\ 3 &amp; 4 \end{pmatrix} = \begin{pmatrix} 11 &amp; 16 \end{pmatrix}$</p><ul><li><p>Multiplication of matrices is not commutative $\rightarrow AB \neq BA$.</p></li></ul>

Express each of the following as a single matrix.

<p>It is given that&nbsp;$X = \begin{pmatrix} a & 9 \\ 3b & 2 \end{pmatrix}$, $Y = \begin{pmatrix} 4 & -1 \\ -5 & -3 \end{pmatrix}$ and $Z = \begin{pmatrix} 9 & 2c \\ 3 & d \end{pmatrix}$. Find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">$3Y$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the values of $a$, $b$, $c$ and $d$ if&nbsp;$X + 3Y = Z$.</span></div></li></ol>

It is given that $X = \begin{pmatrix} a & 9 \\ 3b & 2 \end{pmatrix}$, $Y = \begin{pmatrix} 4 & -1 \\ -5 & -3 \end{pmatrix}$ and $Z = \begin{pmatrix} 9 & 2c \\ 3 & d \end{pmatrix}$. Find

the values of $a$, $b$, $c$ and $d$ if $X + 3Y = Z$.

Multiplication of a Matrix by a Scalar

Isolating Variable Matrices within Scalar-Transformed Systems

<p>If $\begin{pmatrix} 2p \\ -7 \end{pmatrix} - 3 \begin{pmatrix} 1 \\ q \end{pmatrix} = \begin{pmatrix} 9 \\ p - 7 \end{pmatrix}$, find the values of $p$ and $q$.</p>

If $\begin{pmatrix} 2p \\ -7 \end{pmatrix} - 3 \begin{pmatrix} 1 \\ q \end{pmatrix} = \begin{pmatrix} 9 \\ p - 7 \end{pmatrix}$, find the values of $p$ and $q$.

Properties of Scalar-Matrix Products and Matrix Summation

<p>Find the matrix $M$ such that&nbsp;$3M + \begin{pmatrix} 1 & 0 \\ 1 & -1 \end{pmatrix} = \begin{pmatrix} 4 & 3 \\ 7 & -7 \end{pmatrix}$.</p>

Find the matrix $M$ such that $3M + \begin{pmatrix} 1 & 0 \\ 1 & -1 \end{pmatrix} = \begin{pmatrix} 4 & 3 \\ 7 & -7 \end{pmatrix}$.

Fundamentals of Scalar Multiplication in Linear Algebra

<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 650px;"><colgroup><col style="width: 650px;"></colgroup><tbody><tr style="height: 43px;"><td rowspan="2" style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"><p><strong>Tips - Multiplication of a Matrix by a Scalar</strong></p><ul><li><p>When a matrix is multiplied by scalar $k$, the product will be the matrix with each of its elements multiplied by $k$.</p></li></ul><p style="text-align: center;"><br>\[ \text{Example:}\hspace{0.5em} k \begin{pmatrix} e &amp; f \\ g &amp; h \end{pmatrix} = \begin{pmatrix} ke &amp; kf \\ kg &amp; kh \end{pmatrix} \]</p></td></tr><tr style="height: 40px;"></tr></tbody></table></div></div>
<p>If&nbsp;$A = \begin{pmatrix} 0 & 2 \\ 1 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 7 \\ 3 & 5 \end{pmatrix}$, find $B - 3A$.</p>

<p><strong>Tips - Multiplication of a Matrix by a Scalar</strong></p><ul><li><p>When a matrix is multiplied by scalar $k$, the product will be the matrix with each of its elements multiplied by $k$.</p></li></ul><p style="text-align: center;"><br>\[ \text{Example: } k \begin{pmatrix} e &amp; f \\ g &amp; h \end{pmatrix} = \begin{pmatrix} ke &amp; kf \\ kg &amp; kh \end{pmatrix} \]</p>

If $A = \begin{pmatrix} 0 & 2 \\ 1 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 7 \\ 3 & 5 \end{pmatrix}$, find $B - 3A$.

<p>If&nbsp;$A = \begin{pmatrix} 4 & -4 \\ -6 & -1 \end{pmatrix}$, $B = \begin{pmatrix} 3 & -4 \\ 9 & -1 \end{pmatrix}$ and $C = \begin{pmatrix} -8 & 3 \\ 0 & -5 \end{pmatrix}$, find $(A - B) - C$.</p>

If $A = \begin{pmatrix} 4 & -4 \\ -6 & -1 \end{pmatrix}$, $B = \begin{pmatrix} 3 & -4 \\ 9 & -1 \end{pmatrix}$ and $C = \begin{pmatrix} -8 & 3 \\ 0 & -5 \end{pmatrix}$, find $(A - B) - C$.

Addition and Subtraction of Matrices

Unit Mastery: Performing Operations with Multiple Matrices

<p>Given that&nbsp;$X = \begin{pmatrix} 0 & -3 \\ 3 & 0 \end{pmatrix}$ and $Y = \begin{pmatrix} 0 & -1 \\ 2 & 3 \end{pmatrix}$, find the matrix $Z$ such that&nbsp;$X + Z = Y$.</p>

Given that $X = \begin{pmatrix} 0 & -3 \\ 3 & 0 \end{pmatrix}$ and $Y = \begin{pmatrix} 0 & -1 \\ 2 & 3 \end{pmatrix}$, find the matrix $Z$ such that $X + Z = Y$.

Evaluating Linear Combinations of $2 \times 2$ Matrices

<p>Given that&nbsp;$A = \begin{pmatrix} 3 & 1 \\ -8 & 4 \end{pmatrix}$, $B = \begin{pmatrix} -5 & 8 \\ -9 & -6 \end{pmatrix}$ and $C = \begin{pmatrix} -1 & 0 \\ 8 & 3 \end{pmatrix}$, find $C - (A - B)$.</p>

Given that $A = \begin{pmatrix} 3 & 1 \\ -8 & 4 \end{pmatrix}$, $B = \begin{pmatrix} -5 & 8 \\ -9 & -6 \end{pmatrix}$ and $C = \begin{pmatrix} -1 & 0 \\ 8 & 3 \end{pmatrix}$, find $C - (A - B)$.

Math Question Bank