Given the matrices P = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} and Q = \begin{pmatrix… — JAMB UTME

Given the matrices $P = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$ and $Q = \begin{pmatrix} 5 & 0 \\ -2 & 1 \end{pmatrix}$ , what is the product matrix $PQ$ ?

A
$\begin{pmatrix} 10 & 0 \\ -2 & 4 \end{pmatrix}$
$\begin{pmatrix} 4 & 3 \\ -3 & 4 \end{pmatrix}$ Answer
C
$\begin{pmatrix} 10 & 15 \\ -3 & -2 \end{pmatrix}$
D
$\begin{pmatrix} 16 & 3 \\ 13 & 4 \end{pmatrix}$

Answer

The product matrix

PQ

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

To find

PQ

, multiply each row of matrix

P

by each column of matrix

Q

. The entry in row 1, column 1 is

2(5) + 3(-2) = 4

. The entry in row 1, column 2 is

2(0) + 3(1) = 3

. The entry in row 2, column 1 is

1(5) + 4(-2) = -3

. The entry in row 2, column 2 is

1(0) + 4(1) = 4

. Putting these together yields

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

Step-by-Step Solution

Compute the entries of the top row of

PQ

using the first row of

P

and both columns of

Q

First entry:

(2)(5) + (3)(-2) = 10 - 6 = 4

. Second entry:

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

Matrix entry

(1,1)

is the dot product of Row 1 of

P

and Column 1 of

Q

; entry

(1,2)

is the dot product of Row 1 of

P

and Column 2 of

Q

Compute the entries of the bottom row of

PQ

using the second row of

P

and both columns of

Q

Third entry:

(1)(5) + (4)(-2) = 5 - 8 = -3

. Fourth entry:

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

Matrix entry

(2,1)

is the dot product of Row 2 of

P

and Column 1 of

Q

; entry

(2,2)

is the dot product of Row 2 of

P

and Column 2 of

Q

Assemble the computed entries into a

2 \times 2

matrix.

PQ = \begin{pmatrix} 4 & 3 \\ -3 & 4 \end{pmatrix}

Combining row results gives the final product matrix.

Key Concept

Matrix Multiplication (Row-by-Column Dot Product)

Estimated Time:1m 15s

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