Practice Maths

Matrix Operations and Properties

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Key Terms

Matrix addition
Add corresponding entries; defined ONLY for matrices of the same order.
Scalar multiplication
(kA)ij = k × aij; multiply every entry by the scalar k.
Matrix multiplication AB
(AB)ij = dot product of row i of A with column j of B; inner dimensions must match.
Non-commutativity
In general AB ≠ BA; order of factors matters.
Identity matrix I
AI = IA = A for any compatible matrix A.
Transpose rule
(AB)T = BTAT (order reverses when transposing a product).

Matrix Operations — Key Facts

Addition: Only defined for matrices of the same size. Add corresponding entries: (A + B)ij = aij + bij

Scalar multiplication: (kA)ij = k · aij — multiply every entry by the scalar k

Matrix multiplication: AB is defined only when the number of columns of A equals the number of rows of B. The (i, j) entry of AB is the dot product of row i of A with column j of B.

Non-commutativity: In general, AB ≠ BA. Order matters!

Identity matrix: I = [[1,0],[0,1]] satisfies AI = IA = A for any compatible matrix A

Transpose rule: (AB)T = BTAT (order reverses)

Zero matrix: A matrix of all zeros; A + 0 = A

Worked Example 1 — Matrix Multiplication

Let A = [[2, 1], [3, 0]] and B = [[1, 4], [−1, 2]]. Compute AB.

AB entry (1,1): row 1 of A · col 1 of B = 2(1) + 1(−1) = 1

AB entry (1,2): row 1 of A · col 2 of B = 2(4) + 1(2) = 10

AB entry (2,1): row 2 of A · col 1 of B = 3(1) + 0(−1) = 3

AB entry (2,2): row 2 of A · col 2 of B = 3(4) + 0(2) = 12

AB = [[1, 10], [3, 12]]

Worked Example 2 — Showing AB ≠ BA

Using the same A and B above, compute BA and compare with AB.

BA entry (1,1): 1(2) + 4(3) = 14    BA entry (1,2): 1(1) + 4(0) = 1

BA entry (2,1): (−1)(2) + 2(3) = 4    BA entry (2,2): (−1)(1) + 2(0) = −1

BA = [[14, 1], [4, −1]]

Since AB = [[1, 10], [3, 12]] ≠ [[14, 1], [4, −1]] = BA, we confirm AB ≠ BA.

Hot Tip: When multiplying matrices, check dimensions first. An (m × n) matrix times an (n × p) matrix gives an (m × p) matrix. If the inner dimensions don’t match (n ≠ n), the product is undefined. This is the most common source of errors in matrix questions.

What is a Matrix?

A matrix is a rectangular array of numbers arranged in rows and columns. We describe a matrix by its dimensions: an m × n matrix has m rows and n columns. For example, [[2, 1, 3], [4, −1, 0]] is a 2 × 3 matrix. Matrices are a compact and powerful language for expressing systems of linear equations, geometric transformations, and many other mathematical structures that arise throughout science and engineering.

The entry in row i and column j of matrix A is written aij. So for a 2 × 2 matrix A = [[a, b], [c, d]], we have a11 = a, a12 = b, a21 = c, a22 = d. Keeping careful track of row and column indices is essential when computing products.

Addition and Scalar Multiplication

Matrix addition is straightforward: add corresponding entries. This is only possible when both matrices have the same dimensions. If A and B are both 2 × 2 matrices, then (A + B)ij = aij + bij. Addition is commutative (A + B = B + A) and associative — the familiar rules of number arithmetic apply here.

Scalar multiplication is even simpler: multiply every entry by the scalar. So 3A means multiply every entry of A by 3. Combined, addition and scalar multiplication give the matrix analogue of linear combinations, which is fundamental to the study of vector spaces.

Matrix Multiplication: The Dot Product Rule

Matrix multiplication is considerably more subtle than addition. The product AB is formed by taking the dot product of each row of A with each column of B. Specifically, the entry in row i, column j of AB is: (AB)ij = Σk aik bkj.

This means the product AB is only defined when the number of columns of A equals the number of rows of B. If A is m × n and B is n × p, then AB is m × p. Students often forget to check this dimension compatibility condition before attempting to compute a product.

The best method for computing a 2 × 2 product is systematic: write the four entries one at a time, each as a dot product of the relevant row of A with the relevant column of B. Never try to compute “all at once” — work entry by entry.

Non-Commutativity: AB ≠ BA

The most important property that distinguishes matrix multiplication from ordinary arithmetic is that it is not commutative. Even when both AB and BA are defined (e.g., when A and B are both square), they are generally different matrices. This has deep consequences: the order in which you apply matrix transformations matters. Reflecting then rotating gives a different result from rotating then reflecting.

However, matrix multiplication is associative: (AB)C = A(BC). This means you can group products however you like, as long as you preserve the left-to-right order. The distributive laws also hold: A(B + C) = AB + AC and (A + B)C = AC + BC.

The Identity Matrix and Transpose

The identity matrix I (usually 2 × 2 for this course: [[1,0],[0,1]]) plays the role of the number 1 in matrix arithmetic: AI = IA = A for any square matrix A of the same size. It represents the “do nothing” transformation.

The transpose of A, written AT, is formed by swapping rows and columns: the first row becomes the first column, etc. The key transpose rule for products is (AB)T = BTAT — note the reversal of order. This is analogous to the rule (AB)−1 = B−1A−1 for inverses. Both rules arise because the order of operations must be reversed when “undoing” a composition.

Setting Up AX = B

One of the most important uses of matrix multiplication is representing a system of linear equations as a single matrix equation AX = B. For example, the system 2x + y = 5, 3x − 2y = 1 can be written as [[2,1],[3,−2]] [[x],[y]] = [[5],[1]]. This compact form allows us to use matrix inverse theory to solve the system — just as dividing both sides by a in ax = b gives x = b/a, we can multiply both sides by A−1 to get X = A−1B. This connection is explored in the Solving Systems lesson.

Mastery Practice

  1. Fluency

    Q1 — Matrix Addition and Scalar Multiplication

    Let A = [[3, −1], [2, 4]] and B = [[−2, 5], [1, −3]]. Compute: (a) A + B    (b) 3A    (c) 2A − B

  2. Fluency

    Q2 — Matrix Multiplication

    Compute AB where A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]].

  3. Fluency

    Q3 — Multiplying a Matrix by the Identity

    Let A = [[5, −2], [3, 7]] and I = [[1, 0], [0, 1]]. Verify that AI = IA = A.

  4. Fluency

    Q4 — Multiplying Non-Square Matrices

    Let A = [[1, 2, 3], [4, 5, 6]] (2×3) and B = [[7], [8], [9]] (3×1). Compute AB and state its dimensions.

  5. Understanding

    Q5 — Showing AB ≠ BA

    Let A = [[1, 2], [0, 1]] and B = [[1, 0], [3, 1]]. Compute both AB and BA and confirm that AB ≠ BA.

  6. Understanding

    Q6 — Finding Unknown Entries

    Find x and y given that [[x, 2], [1, y]] × [[3], [4]] = [[11], [7]].

  7. Understanding

    Q7 — Transpose Property

    Let A = [[1, 3], [2, 4]] and B = [[0, 1], [2, 5]]. Verify that (AB)T = BTAT.

  8. Understanding

    Q8 — Setting Up a System as AX = B

    Write the system 3x + 2y = 7, −x + 4y = 5 in matrix form AX = B. Identify A, X, and B.

  9. Problem Solving

    Q9 — Powers of a Matrix

    Let A = [[1, 1], [0, 1]]. Compute A², A³, and conjecture a formula for An.

  10. Problem Solving

    Q10 — Proof: (A + B)² ≠ A² + 2AB + B² in General

    Using A = [[1, 1], [0, 0]] and B = [[0, 0], [1, 1]], show that (A + B)² ≠ A² + 2AB + B², and explain why this “expansion” fails for matrices.

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