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<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 \(r\).

Multiplication of a Matrix by a Scalar

Multiplication of Two Matrices

Equality of Matrices

Matrices

<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 \(k\).

Addition and Subtraction of Matrices

<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>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 \(X + 3Y = Z\).

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 \(B - 3A\).

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