Find the product matrix AB given that A = \begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix} and… — JAMB UTME

Find the product matrix $AB$ given that $A = \begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 4 & 5 \\ 1 & 2 \end{pmatrix}$ .

\begin{pmatrix} 6 & 9 \\ 3 & 6 \end{pmatrix}

The matrix product

AB

is found by performing row-by-column multiplication. Computing row 1 gives

(1)(4)+(2)(1) = 6

and

(1)(5)+(2)(2) = 9

. Computing row 2 gives

(0)(4)+(3)(1) = 3

and

(0)(5)+(3)(2) = 6

. Thus, the resulting matrix is

\begin{pmatrix} 6 & 9 \\ 3 & 6 \end{pmatrix}

Set up row-by-column matrix multiplication formula for

2 \times 2

matrices.

The entry at row

i

, column

j

of product

AB

is the dot product of row

i

A

and column

j

B

Matrix multiplication requires combining rows of the left matrix with columns of the right matrix.

Calculate row 1 entries of the product matrix.

First element:

(1)(4) + (2)(1) = 4 + 2 = 6

. Second element:

(1)(5) + (2)(2) = 5 + 4 = 9

Multiply elements of row 1 of

A

by corresponding elements of columns 1 and 2 of

B

and sum them.

Calculate row 2 entries of the product matrix.

Third element:

(0)(4) + (3)(1) = 0 + 3 = 3

. Fourth element:

(0)(5) + (3)(2) = 0 + 6 = 6

Multiply elements of row 2 of

A

by corresponding elements of columns 1 and 2 of

B

and sum them.

Assemble the product matrix.

AB = \begin{pmatrix} 6 & 9 \\ 3 & 6 \end{pmatrix}

Combine all computed row and column entries into a single

2 \times 2

matrix.

Matrix Multiplication (

2 \times 2

Matrix Product)

Question