Operations on Matrices – Matrices | Class 12 Maths

Understand

Addition and subtraction are done entry by entry and need matrices of the same order. Scalar multiplication multiplies every entry by the scalar: \(kA = [k\,a_{ij}]\).

Multiplication

If \(A\) is \(m\times n\) and \(B\) is \(n\times p\), the product \(AB\) is the \(m\times p\) matrix whose \((i, j)\) entry is

\[(AB)_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} + \dots + a_{in}b_{nj}\]

Each entry of AB is a row-by-column product.

For example, \(\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\begin{bmatrix} 5 \\ 6 \end{bmatrix} = \begin{bmatrix} 1\cdot5+2\cdot6 \\ 3\cdot5+4\cdot6 \end{bmatrix} = \begin{bmatrix} 17 \\ 39 \end{bmatrix}\).

Important properties

Associative: \((AB)C = A(BC)\); distributive: \(A(B+C) = AB + AC\).
Not commutative: \(AB\) and \(BA\) are usually different (one may not even exist).
\(AB = O\) does not imply \(A = O\) or \(B = O\).
\(AI = IA = A\) for a square matrix \(A\).

Key Concepts

Conformability: columns of the first = rows of the second.
Row × column rule for each entry.
Order of factors matters.

Formula Bank

Entry of a product

\[(AB)_{ij} = \sum_{k=1}^{n} a_{ik}\,b_{kj}\]

Multiply row \(i\) of \(A\) by column \(j\) of \(B\) term by term and add. Requires columns of \(A\) = rows of \(B\).

Order of a product

\[(m\times n)(n\times p) = (m\times p)\]

The inner dimensions must match; the outer ones give the order of the product.

Key Points

AB ≠ BA in general

Never assume matrices commute. Keep the order of factors in every step.

Even when both \(AB\) and \(BA\) exist, they may differ – and one may exist while the other does not.

No cancellation law

\(AB = AC\) does not imply \(B = C\), and \(AB = O\) does not imply \(A = O\) or \(B = O\).

Common Mistakes

Squaring entries instead of the matrix

Writing \(\begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix}^2 = \begin{bmatrix} 4 & 1 \\ 0 & 9 \end{bmatrix}\).

Correct: \(A^2 = A\cdot A = \begin{bmatrix} 4 & 5 \\ 0 & 9 \end{bmatrix}\). Matrix powers use row × column multiplication.

Using (A + B)² = A² + 2AB + B²

Applying the number identity to matrices.

Correct: \((A+B)^2 = A^2 + AB + BA + B^2\). It reduces to the familiar form only if \(AB = BA\).

Treating matrix algebra like numbers

\(AB=O\) does not force \(A=O\) or \(B=O\), and \((A+B)^2=A^2+AB+BA+B^2\), not \(A^2+2AB+B^2\), unless \(AB=BA\).

Solved Examples

Checking whether AB = BA

For \(A = \begin{bmatrix} 1 & -2 \\ 3 & 0 \end{bmatrix}\) and \(B = \begin{bmatrix} 2 & 1 \\ -1 & 4 \end{bmatrix}\), find \(AB\) and \(BA\). Are they equal?

\(AB = \begin{bmatrix} 4 & -7 \\ 6 & 3 \end{bmatrix}\).
\(BA = \begin{bmatrix} 5 & -4 \\ 11 & 2 \end{bmatrix}\).
Compare corresponding entries: they differ.

Final Answer: \(AB \ne BA\): matrix multiplication is not commutative.

Practice & Topic Test

Topic Practice

A fresh set of questions from this topic. Change answers freely – solutions appear after you submit.

Start Practice

Topic Test

Timed: up to 10 questions in 15 minutes.

Start Test

Topic Summary

Add/subtract entry-wise (same order). \(kA\) scales every entry. \((m\times n)(n\times p) = m\times p\), entries by row × column. \(AB \ne BA\) in general.