Q.Find the value of $a, b, c$ and $d$ from the equation: $\begin{bmatrix} a-b & 2a+c \\ 2a-b & 3c+d \end{bmatrix} = \begin{bmatrix} -1 & 5 \\ 0 & 13 \end{bmatrix}$

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

Step 1: Equate the entries in position (1,1):

$a - b = -1$

Step 2: Equate (1,2):

$2a + c = 5$

Step 3: Equate (2,1):

$2a - b = 0$

Step 4: Equate (2,2):

$3c + d = 13$

Now solve the system. From $2a - b = 0$, we get $b = 2a$. Substitute into $a - b = -1$: …

When two matrices are written as equal, every entry in the same position must be identical. This is the definition of matrix equality — no shortcuts, no tricks. So the given $2\times 2$ matrix equation is really a compact way of writing four separate equations.

Let’s unpack them.

1. Equate the (1,1) entries (top-left):

   $a - b = -1$

   This is our first equation.

2. Equate the (1,2) entries (top-right):

   $2a + c = 5$

   Second equation.

3. Equate the (2,1) entries (bottom-left):

   $2a - b = 0$

   Third equation.

4. Equate the (2,2) entries (bottom-right):

   $3c + d = 13$

   Fourth equation.

Now we have a system:

Notice that the first and third equations both involve $a$ and $b$ only. Solve those together first.

From $2a - b = 0$, we get $b = 2a$.

Substitute into $a - b = -1$:

$a - 2a = -1 \implies -a = -1 \implies a = 1$.

Then $b = 2(1) = 2$.

Now $a$ is known. Use the second equation: $2a + c = 5 \implies 2(1) + c = 5 \implies c = 3$. …

Method: Solving for Unknowns From a Matrix Equation

Use this whenever two matrices with unknown entries are set equal. Matrix equality turns the single equation into a system of scalar equations, one per position.

Steps

Step 1: Equate corresponding entries.

Two matrices are equal iff they have the same order and $a_{ij}=b_{ij}$ for every position. Write one equation for each cell — a $2\times2$ equation gives four equations.

Step 2: Group and solve the coupled equations. …

Common Mistakes

Mistake 1: Using only some entries instead of all of them.

Why it's wrong: each of the four positions is a separate constraint; ignoring one can give a wrong or incomplete answer. Correct approach: form and use one equation per entry.

Mistake 2: Sign slips with negative right-hand values. …