Given the matrices A = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix} and B = \begin{pmatrix… — JAMB UTME

Given the matrices $A = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 4 \\ 2 & -1 \end{pmatrix}$ , what is the product matrix $AB$ ?

$\begin{pmatrix} 4 & 7 \\ 6 & -3 \end{pmatrix}$ Answer
B
$\begin{pmatrix} 2 & 4 \\ 0 & -3 \end{pmatrix}$
C
$\begin{pmatrix} 2 & 13 \\ 4 & -1 \end{pmatrix}$
D
$\begin{pmatrix} 4 & 9 \\ 6 & -3 \end{pmatrix}$

Answer

The matrix product

AB

\begin{pmatrix} 4 & 7 \\ 6 & -3 \end{pmatrix}

To find

AB

, each element

c_{ij}

is formed by taking the dot product of row

i

of matrix

A

and column

j

of matrix

B

. Performing these steps gives

c_{11} = 2(1) + 1(2) = 4

c_{12} = 2(4) + 1(-1) = 7

c_{21} = 0(1) + 3(2) = 6

, and

c_{22} = 0(4) + 3(-1) = -3

, yielding

\begin{pmatrix} 4 & 7 \\ 6 & -3 \end{pmatrix}

Step-by-Step Solution

Multiply the first row of

A

by the first column of

B

to find element

(1,1)

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

Matrix multiplication rule requires taking the dot product of rows from the first matrix and columns from the second matrix.

Multiply the first row of

A

by the second column of

B

to find element

(1,2)

2(4) + 1(-1) = 8 - 1 = 7

Evaluates the element in row 1, column 2 of the resulting matrix.

Multiply the second row of

A

by the first column of

B

to find element

(2,1)

0(1) + 3(2) = 0 + 6 = 6

Evaluates the element in row 2, column 1 of the resulting matrix.

Multiply the second row of

A

by the second column of

B

to find element

(2,2)

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

Evaluates the element in row 2, column 2 of the resulting matrix.

Key Concept

Matrix Multiplication

Estimated Time:1m 0s

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