4.5 Multiplication of Two Matrices (Sample Question 1)

Posted on April 24, 2020 by user

Example 1:
Find the product of the following pairs of matrices.

\begin{array}{l} (a) (1 5 2) (\begin{array}{l} 2 \\ 4 \\ 3 \end{array}) \\ (b) (\begin{matrix} 2 & 8 \\ - 3 & 1 \end{matrix}) (\begin{matrix} 1 & 0 \\ 4 & - 2 \end{matrix}) \\ (c) (\begin{array}{l} - 3 \\ 5 \end{array}) (2 6) \\ (d) (\begin{matrix} 0 & 4 \\ - 1 & 3 \end{matrix}) (\begin{array}{l} 7 \\ - 2 \end{array}) \\ (e) (7 - 4) (\begin{matrix} - 2 & 0 \\ - 1 & 3 \end{matrix}) \end{array}

Solution:

\begin{array}{l} (a) \\ (1 5 2) (\begin{array}{l} 2 \\ 4 \\ 3 \end{array}) \leftarrow \begin{array}{l} Matrices analysis \\ 1 \times 3 and 3 \times 1 \\ ↓ ↓ \\ = 1 \times 1 matrix \end{array} \\ = (1 \times 2 \oplus 5 \times 4 \oplus 2 \times 3) \\ = (2 + 20 + 6) \\ = (28) \end{array}

(b)

\begin{array}{l} (\begin{matrix} 2 & 8 \\ - 3 & 1 \end{matrix}) (\begin{matrix} 1 & 0 \\ 4 & - 2 \end{matrix}) \leftarrow \begin{array}{l} Matrices analysis \\ 2 \times 2 and 2 \times 2 \\ ↓ ↓ \\ = 2 \times 2 matrix \end{array} \\ = (\begin{matrix} 2 \times 1 + 8 \times 4 & 2 \times 0 + 8 \times - 2 \\ - 3 \times 1 + 1 \times 4 & - 3 \times 0 + 1 \times - 2 \end{matrix}) \\ = (\begin{matrix} 34 & - 16 \\ 1 & - 2 \end{matrix}) \end{array}

(c)

\begin{array}{l} (\begin{array}{l} - 3 \\ 5 \end{array}) (2 6) \leftarrow \begin{array}{l} Matrices analysis \\ 2 \times 1 and 1 \times 2 \\ ↓ ↓ \\ = 2 \times 2 matrix \end{array} \\ = (\begin{matrix} - 3 \times 2 & - 3 \times 6 \\ 5 \times 2 & 5 \times 6 \end{matrix}) \\ = (\begin{matrix} - 6 & - 18 \\ 10 & 30 \end{matrix}) \end{array}

(d)

\begin{array}{l} (\begin{matrix} 0 & 4 \\ - 1 & 3 \end{matrix}) (\begin{array}{l} 7 \\ - 2 \end{array}) \leftarrow \begin{array}{l} Matrices analysis \\ 2 \times 2 and 2 \times 1 \\ ↓ ↓ \\ = 2 \times 1 matrix \end{array} \\ = (\begin{array}{l} 0 \times 7 + 4 \times - 2 \\ - 1 \times 7 + 3 \times - 2 \end{array}) \\ = (\begin{array}{l} - 8 \\ - 13 \end{array}) \end{array}

(e)

\begin{array}{l} (7 - 4) (\begin{matrix} - 2 & 0 \\ - 1 & 3 \end{matrix}) \leftarrow \begin{array}{l} Matrices analysis \\ 1 \times 2 and 2 \times 2 \\ ↓ ↓ \\ = 1 \times 2 matrix \end{array} \\ = (7 \times - 2 + (- 4 \times - 1) 7 \times 0 + (- 4 \times 3)) \\ = (- 14 + 4 0 - 12) \\ = (- 10 - 12) \end{array}