Q.Find the values of $a, b, c,$ and $d$ from the following equation: $\begin{bmatrix} 2a+b & a-2b \\ 5c-d & 4c+3d \end{bmatrix} = \begin{bmatrix} 4 & -3 \\ 11 & 24 \end{bmatrix}$

Concept: Vector Equality — two matrices are equal if and only if their corresponding entries are equal.

Step 1: Equate the (1,1) and (1,2) entries:

Step 2: Solve for $a$ and $b$. From the second equation, $a = 2b - 3$. Substitute into the first:

Then $a = 2(2) - 3 = 1$.

Step 3: Equate the (2,1) and (2,2) entries:

$5c - d = 11, \quad 4c + 3d = 24$ …

The core idea here is vector equality applied to matrices. When two matrices are declared equal, every entry in the same position must match exactly. That’s not a suggestion — it’s the definition. So this single matrix equation is really a compact way of writing four separate equations, one for each position.

Let’s unpack it.

We have:

Since the matrices are $2 \times 2$, equality means:

1. Top-left entry: $2a + b = 4$
2. Top-right entry: $a - 2b = -3$
3. Bottom-left entry: $5c - d = 11$
4. Bottom-right entry: $4c + 3d = 24$

Notice something beautiful: the equations for $a, b$ are completely separate from those for $c, d$. They don’t mix. So we can solve them as two independent pairs.

Solving for $a$ and $b$

We have:

From (2), we get $a = 2b - 3$. Substitute into (1):

Then $a = 2(2) - 3 = 1$. …

Method: Turning a matrix equality into simultaneous equations

Use this when equal matrices have entries that are linear combinations of unknowns, giving small systems to solve.

Steps

Step 1: Equate corresponding entries.

Each position gives one linear equation in the unknowns.

Step 2: Group the equations that share unknowns. …

Common Mistakes

Mistake 1: Pairing the wrong constant with an entry.

Why it's wrong: the bottom-right entry gives $4c + 3d = 24$, not $= 11$; the numbers $11$ and $24$ sit in different positions. Correct approach: match each expression to the number in the same position.

Mistake 2: Arithmetic slips in elimination. …