Question 1, Exercise 2.1

Solutions of Question 1 of Exercise 2.1 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.

Express as a single matrix $$\left[ \begin{matrix} 1 & 2 & 4 \\ \end{matrix} \right] \left[ \begin{matrix} 1 & 0 & 2 \\ 2 & 0 & 1 \\ 0 & 1 & 2 \\ \end{matrix} \right] \left[ \begin{matrix} 2 \\ 4 \\ 6 \\ \end{matrix} \right]$$

\begin{align}&\left[ \begin{matrix} 1 & 2 & 4 \\ \end{matrix} \right]\left[ \begin{matrix} 1 & 0 & 2 \\ 2 & 0 & 1 \\ 0 & 1 & 2 \\ \end{matrix} \right]\left[ \begin{matrix} 2 \\ 4 \\ 6 \\ \end{matrix} \right] \\ &=\left[ \begin{matrix} 1 & 2 & 4 \\ \end{matrix} \right]\left[ \begin{matrix} 2+0+12 \\ 4+0+6 \\ 0+4+12 \\ \end{matrix} \right] \\ &=\left[ \begin{matrix} 1 & 2 & 4 \\ \end{matrix} \right]\left[ \begin{matrix} 14 \\ 10 \\ 16 \\ \end{matrix} \right]\\ &=\left[ 14+20+64 \right]\\ &=[98].\end{align}

Express as a single matrix $$\left[ \begin{matrix} 1 & -2 & 3 \\ \end{matrix} \right]\left[ \begin{matrix} 2 & -1 & 5 \\ 0 & 2 & 4 \\ -7 & 5 & 0 \\ \end{matrix} \right]-\left[ \begin{matrix} 2 & -5 & 7 \\ \end{matrix} \right]$$

\begin{align}&\left[ \begin{matrix} 1 & -2 & 3 \\ \end{matrix} \right]\left[ \begin{matrix} 2 & -1 & 5 \\ 0 & 2 & 4 \\ -7 & 5 & 0 \\ \end{matrix} \right]-\left[ \begin{matrix} 2 & -5 & 7 \\ \end{matrix} \right]\\ &=\left[ \begin{matrix} 2+0-21 & -1-4+15 & 5-8+0 \\ \end{matrix} \right]-\left[ \begin{matrix} 2 & -5 & 7 \\ \end{matrix} \right]\\ &=\left[ \begin{matrix} -19 & 10 & -3 \\ \end{matrix} \right]-\left[ \begin{matrix} 2 & -5 & 7 \\ \end{matrix} \right]\\ &=\left[ \begin{matrix} -19-2 & 10+5 & -3-7 \\ \end{matrix} \right]\\ &=\left[ \begin{matrix} -21 & 15 & -10 \\ \end{matrix} \right] \end{align}

Express as a single matrix $$\left[ \begin{matrix} 7 & 1 & 2 \\ 9 & 2 & 1 \\ \end{matrix} \right]\left[ \begin{matrix} 3 \\ 4 \\ 5 \\ \end{matrix} \right]+2\left[ \begin{matrix} 4 \\ 2 \\ \end{matrix} \right]$$

\begin{align}&\left[ \begin{matrix} 7 & 1 & 2 \\ 9 & 2 & 1 \\ \end{matrix} \right]\left[ \begin{matrix} 3 \\ 4 \\ 5 \\ \end{matrix} \right]+2\left[ \begin{matrix} 4 \\ 2 \\ \end{matrix} \right] \\ &=\left[ \begin{matrix} 21+4+10 \\ 27+8+5 \\ \end{matrix} \right]+\left[ \begin{matrix} 8 \\ 4 \\ \end{matrix} \right] \\ &=\left[ \begin{matrix} 35 \\ 40 \\ \end{matrix} \right]+\left[ \begin{matrix} 8 \\ 4 \\ \end{matrix} \right]\\ &=\left[ \begin{matrix} 35+8 \\ 40+4 \\ \end{matrix} \right]\\ &=\left[ \begin{matrix} 43 \\ 44 \\ \end{matrix} \right] \end{align}

Express as a single matrix $$\left\{ \left[ \begin{matrix} 1 & 3 \\ -1 & -4 \\ \end{matrix} \right]+\left[\begin{matrix} 3 & -2 \\ -1 & 1 \\ \end{matrix} \right] \right\}\left[ \begin{matrix} 1 & 3 & 5 \\ 2 & 4 & 6 \\ \end{matrix} \right]$$

\begin{align}&\left\{ \begin{bmatrix} 1 & 3 \\ -1 & -4 \end{bmatrix}+\begin{bmatrix}3 & -2 \\-1 & 1\end{bmatrix} \right\} \begin{bmatrix}1 & 3 & 5\\ 2 & 4 & 6 \end{bmatrix} \\ &=\begin{bmatrix}1+3 & 3-2 \\ -1-1 & -4+1 \end{bmatrix} \begin{bmatrix}1 & 3 & 5 \\2 & 4 & 6 \end{bmatrix}\\ &=\begin{bmatrix}4 & 1 \\-2 & -3 \end{bmatrix}\begin{bmatrix}1 & 3 & 5 \\2 & 4 & 6 \end{bmatrix} \\ &=\begin{bmatrix}4+2 & 12+4 & 20+6 \\-2-6 & -6-12 & -10-18 \end{bmatrix} \\ &=\begin{bmatrix}6 & 16 & 26 \\-8 & -18 & -28 \end{bmatrix} \end{align}

Question 2 >