# Equal Matrices (Sample Questions)

Example 1:
State the values of the unknowns in the following pairs of equal matrix.
$\left(\begin{array}{cc}1& x+2\\ 4-y& -1\end{array}\right)=\left(\begin{array}{cc}1& 3\\ 2& -1\end{array}\right)$

Solution:

$\left(\begin{array}{cc}1& x+2\\ 4-y& -1\end{array}\right)=\left(\begin{array}{cc}1& 3\\ 2& -1\end{array}\right)$

x + 2 = 3

x = 1

4 – y = 2
y = –2
y = 2

Example 2:
Calculate the values of p and q in each of the following matrix equations.
$\begin{array}{l}\text{(a)}\left(\begin{array}{cc}3& 2p+q\\ p& -3\end{array}\right)=\left(\begin{array}{cc}3& 1\\ 8-2q& -3\end{array}\right)\\ \text{(b)}\left(\begin{array}{cc}10& 0\\ 5p-8& 1\end{array}\right)=\left(\begin{array}{cc}p-2q& 0\\ -4q& 1\end{array}\right)\end{array}$

Solution:
$\text{(a)}\left(\begin{array}{cc}3& 2p+q\\ p& -3\end{array}\right)=\left(\begin{array}{cc}3& 1\\ 8-2q& -3\end{array}\right)$

2p + q = 1
q = 1 – 2p —-(1)
p = 8 – 2q —-(2)

Substitute (1) into (2),
p = 8 – 2(1 – 2p)
p = 8 – 2 + 4p
p – 4p = 6
–3p = 6
p = –2

Substitute p = 2 into (1),
q = 1 – 2(–2)
q = 5

$\text{(b)}\left(\begin{array}{cc}10& 0\\ 5p-8& 1\end{array}\right)=\left(\begin{array}{cc}p-2q& 0\\ -4q& 1\end{array}\right)$

10 = p – 2q
p = 10 + 2q —-(1)
5p – 8 = –4q —-(2)

Substitute (1) into (2),
5 (10 + 2q) – 8 = –4q
50 + 10q – 8 = –4q
14q = –42
q = –3

Substitute q = –3 into (1),
p = 10 + 2(–3)
p = 4