Find the value of $a, b, c$, and $d$ from the equation:

Solution:

$\left[\begin{array}{ll}a-b & 2 a+c \\ 2 a-b & 3 c+d\end{array}\right]=\left[\begin{array}{ll}-1 & 5 \\ 0 & 13\end{array}\right]$

As the two matrices are equal, their corresponding elements are also equal.

Comparing the corresponding elements, we get:

a − b = −1 … (1)

2a − b = 0 … (2)

2a + c = 5 … (3)

 

3c + d = 13 … (4)

From (2), we have:

b = 2a

 

Then, from (1), we have:

$a-2 a=-1$

$\Rightarrow a=1$

$\Rightarrow b=2$

Now, from (3), we have:

2 ×1 + c = 5

$\Rightarrow c=3$

From (4) we have:

$3 \times 3+d=13$

$\Rightarrow 9+d=13 \Rightarrow d=4$

$\therefore a=1, b=2, c=3$, and $d=4$