Operations on Matrices> Matrix Multiplication

Matrix multiplication is a fundamental operation in algebra that helps us represent and solve systems of equations, perform coordinate transformations, and model real-life situations like network flows and data transformations.

Let \[A = [a_{ij}]\] be an \[m \times n\] matrix and \[B = [b_{jk}]\] be an \[n \times p\] matrix.

Then the product C = AB is an \[m \times p\] matrix \[C = [c_{ik}]\], where each entry \[c_{ik}\] is given by:

If \[\mathrm{A=} \begin{bmatrix} 0 & 6 & 7 \\ -6 & 0 & 8 \\ 7 & -8 & 0 \end{bmatrix},\mathrm{B=} \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 2 \\ 1 & 2 & 0 \end{bmatrix},\mathrm{C=} \begin{bmatrix} 2 \\ -2 \\ 3 \end{bmatrix}\]

Calculate AC, BC and (A + B)C. Also, verify that (A + B)C = AC + BC

Solution: Now, \[\mathbf{A}+\mathbf{B}= \begin{bmatrix} 0 & 7 & 8 \\ -5 & 0 & 10 \\ 8 & -6 & 0 \end{bmatrix}\]

So \[\mathrm{(A+B)C}= \begin{bmatrix} 0 & 7 & 8 \\ -5 & 0 & 10 \\ 8 & -6 & 0 \end{bmatrix} \begin{bmatrix} 2 \\ -2 \\ 3 \end{bmatrix}= \begin{bmatrix} 0-14+24 \\ -10+0+30 \\ 16+12+0 \end{bmatrix}= \begin{bmatrix} 10 \\ 20 \\ 28 \end{bmatrix}\]

Further \[\mathrm{AC}= \begin{bmatrix} 0 & 6 & 7 \\ -6 & 0 & 8 \\ 7 & -8 & 0 \end{bmatrix} \begin{bmatrix} 2 \\ -2 \\ 3 \end{bmatrix}= \begin{bmatrix} 0-12+21 \\ -12+0+24 \\ 14+16+0 \end{bmatrix}= \begin{bmatrix} 9 \\ 12 \\ 30 \end{bmatrix}\]

and \[\mathrm{BC}= \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 2 \\ 1 & 2 & 0 \end{bmatrix} \begin{bmatrix} 2 \\ -2 \\ 3 \end{bmatrix}= \begin{bmatrix} 0-2+3 \\ 2+0+6 \\ 2-4+0 \end{bmatrix}= \begin{bmatrix} 1 \\ 8 \\ -2 \end{bmatrix}\]

So \[\mathrm{AC}+\mathrm{BC}= \begin{bmatrix} 9 \\ 12 \\ 30 \end{bmatrix}+ \begin{bmatrix} 1 \\ 8 \\ -2 \end{bmatrix}= \begin{bmatrix} 10 \\ 20 \\ 28 \end{bmatrix}\]

Clearly, (A + B) C = AC + BC

• Matrix multiplication is row-by-column, not term-wise.

• Product AB exists only if columns of A = rows of B.

• If A is \[m \times n\] and B is \[n \times p\], then AB is \[m \times p\].

• In general, \[AB \neq BA\], and sometimes one product may not even be defined.

• Matrix multiplication is associative and distributive over addition.

• Identity matrix acts as a multiplicative identity: AI = IA = A.

• Zero matrix absorbs multiplication: AO = OA = O.