Q.If $\begin{bmatrix} x+3 & z+4 & 2y-7 \\ -6 & a-1 & 0 \\ b-3 & -21 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 6 & 3y-2 \\ -6 & -3 & 2c+2 \\ 2b+4 & -21 & 0 \end{bmatrix}$ Find the values of $a, b, c, x, y$ and $z$.

Idea: Two matrices are equal iff corresponding entries are equal, so read off one equation per position.

• $(1,1):\ x+3 = 0 \Rightarrow x = -3$
• $(1,2):\ z+4 = 6 \Rightarrow z = 2$
• $(1,3):\ 2y-7 = 3y-2 \Rightarrow -5 = y \Rightarrow y = -5$
• $(2,2):\ a-1 = -3 \Rightarrow a = -2$
• $(2,3):\ 0 = 2c+2 \Rightarrow c = -1$
• $(3,1):\ b-3 = 2b+4 \Rightarrow -7 = b \Rightarrow b = -7$

The remaining positions ($-6=-6$, $-21=-21$, $0=0$) are automatically satisfied.

Two matrices of the same order are equal exactly when every entry in the same position matches. So this single matrix equation splits into nine ordinary equations; six carry the unknowns and three are automatically true.

Read off each position

1. $(1,1):\ x+3 = 0 \Rightarrow x = -3$.
2. $(1,2):\ z+4 = 6 \Rightarrow z = 2$.
3. $(1,3):\ 2y-7 = 3y-2$. Bring terms together: $-7+2 = 3y-2y$, so $-5 = y$, i.e. $y = -5$.
4. $(2,1):\ -6 = -6$ — always true.
5. $(2,2):\ a-1 = -3 \Rightarrow a = -2$.
6. $(2,3):\ 0 = 2c+2 \Rightarrow 2c = -2 \Rightarrow c = -1$.
7. $(3,1):\ b-3 = 2b+4$. Then $-3-4 = 2b-b$, so $b = -7$.
8. $(3,2):\ -21 = -21$ — always true.
9. $(3,3):\ 0 = 0$ — always true.

All nine equations are consistent, so every unknown is determined.

Method: Solving unknowns from equality of two matrices

Use this whenever two matrices are set equal and you must find the unknowns inside them.

Steps

Step 1: Use the equality condition.

Two matrices of the same order are equal iff every corresponding entry is equal. This turns one matrix equation into a set of scalar equations, one per position.

Step 2: Write down each entry equation.

Match position by position. Some positions give trivially true statements (e.g. $-6 = -6$) and can be skipped; the rest are equations in the unknowns.

Step 3: Solve each equation, watching the signs.

Many are one-line linear equations; isolate each unknown and solve.

Common Mistakes

Mistake 1: Sign errors when rearranging.

Why it's wrong: $2y - 7 = 3y - 2$ gives $y = -5$ (not $+5$), and $b - 3 = 2b + 4$ gives $b = -7$. Correct approach: move variables to one side and constants to the other, tracking each sign.

Mistake 2: Matching entries in the wrong positions.

Why it's wrong: equality is position-by-position; comparing $(1,3)$ with $(3,1)$ produces false equations. Correct approach: equate only entries in identical (row, column) positions.

Mistake 3: Assuming an unknown appears where it doesn't.

Why it's wrong: some positions are pure constants and give no information about the unknowns. Correct approach: extract equations only from positions that actually contain an unknown.