Matrix multiplication method: rows × columns with worked example.

Multiply Two 2×2 Matrices — Step by Step

Question

Find the product AB where:

A = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 5 & 1 \\ 2 & 3 \end{pmatrix}

Solution — Step by Step

For matrix multiplication AB, the number of columns in A must equal the number of rows in B. Here A is 2×2 and B is 2×2 — columns of A = 2 = rows of B. We’re good to proceed.

The product of a 2×2 matrix with a 2×2 matrix gives a 2×2 matrix. Label the result:

AB = \begin{pmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{pmatrix}

Each entry $c_{ij}$ comes from the i-th row of A dotted with the j-th column of B.

Row 1 of A is [2, 3]. We pair it with each column of B.

c_{11} = (2)(5) + (3)(2) = 10 + 6 = 16

c_{12} = (2)(1) + (3)(3) = 2 + 9 = 11

Row 2 of A is [1, 4]. Same process with each column of B.

c_{21} = (1)(5) + (4)(2) = 5 + 8 = 13

c_{22} = (1)(1) + (4)(3) = 1 + 12 = 13

\boxed{AB = \begin{pmatrix} 16 & 11 \\ 13 & 13 \end{pmatrix}}

Why This Works

The row-times-column rule isn’t arbitrary — it comes from how linear transformations compose. When we apply transformation B first, then A, the combined effect on any vector is captured exactly by AB. Each entry $c_{ij}$ measures how much the i-th output component depends on the j-th input component after both transformations.

Think of it this way: $c_{11} = 16$ tells us the top-left entry of the result captures contributions from every element in row 1 of A interacting with column 1 of B. The dot product is doing exactly that — summing up all those interactions.

This is why matrix multiplication is defined the way it is, even if it looks mechanical at first. Once you see matrices as transformation machines, the definition feels inevitable.

Alternative Method

For 2×2 matrices specifically, some students prefer a visual overlay method. Write A to the left and B on top, then each cell of the result sits at the intersection:

        [5  1]    [2  3]
        [2  3]

Row 1 × Col 1: 2·5 + 3·2 = 16   →  top-left
Row 1 × Col 2: 2·1 + 3·3 = 11   →  top-right
Row 2 × Col 1: 1·5 + 4·2 = 13   →  bottom-left
Row 2 × Col 2: 1·1 + 4·3 = 13   →  bottom-right

Same arithmetic, just laid out differently. Use whichever helps you track which row-column pair you’re working on — losing track mid-calculation is the most common error in board exams.

In JEE Main, matrix multiplication questions often test whether you know AB ≠ BA. Always check which order the question specifies. Computing BA for the matrices above gives a completely different result — this non-commutativity is a favourite trick in MCQ options.

Common Mistake

The classic error: multiplying entry by entry like scalar multiplication. Students write $c_{11} = 2 \times 5 = 10$ and stop there, forgetting to add the second product $3 \times 2 = 6$ . This gives the Hadamard product (element-wise), not the matrix product. In NCERT exercises and board papers, this mistake costs full marks because every single entry will be wrong.

Always remember: each entry in AB requires a sum of products, not just one product. Two terms for 2×2, three terms for 3×3, and so on.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

Try These Next