Q.Find values of $a$ and $b$ if $A = B$, where $A = \begin{bmatrix} a+4 & 3b \\ 8 & -6 \end{bmatrix}$, $B = \begin{bmatrix} 2a+2 & b^2+2 \\ 8 & b^2-5b \end{bmatrix}$.

Two matrices are equal exactly when all corresponding entries are equal. Equate the entries of $A$ and $B$.

(1,1): $a+4=2a+2\ \Rightarrow\ a=2$.

(1,2): $3b=b^2+2\ \Rightarrow\ b^2-3b+2=0\ \Rightarrow\ (b-1)(b-2)=0$, so $b=1$ or $b=2$. …

The idea

Two matrices of the same order are equal precisely when every corresponding entry matches. So $A=B$ turns into four scalar equations — one per position. We solve them and keep only the values that satisfy all of them.

Entry by entry

1. Position (1,1):

2. Position (1,2):

so $b=1$ or $b=2$.

3. Position (2,1): $8=8$ — always true, no information.

4. Position (2,2):

so $b=2$ or $b=3$. …

Method: Finding unknowns from a matrix equality

Use this when two matrices are declared equal and you must find unknown letters inside them.

Steps

Step 1: Apply the equality condition.

Two matrices are equal only if they have the same order and every corresponding entry is equal. This converts one matrix equation into several scalar equations — one per position.

Step 2: Write and solve each scalar equation. …

Common Mistakes

Mistake 1: Accepting a value that fits only one entry.

Why it's wrong: the $(1,2)$ entry allows $b=1$ or $b=2$, but the $(2,2)$ entry allows $b=2$ or $b=3$; only $b=2$ satisfies both. Correct approach: take the value common to all equations an unknown appears in.

Mistake 2: Skipping the $(2,2)$ equation. …