Higher Mathematics 1st Paper · Classes 11-12 · Chapter 1 · 1.2

Multiplication of Matrices and Powers.

You multiply A by B and get AB. Then you ask for BA and the book says it cannot be found. That is not your mistake. With matrices you check the orders before you multiply a single number.

1. Will the product exist, and what order?

The number of columns of A must equal the number of rows of B before AB can be found. If A is m × n and B is n × r, then AB has order m × r.

Even when AB exists, BA may not. For BA the number of columns of B, which is r, must equal the number of rows of A, which is m. If they differ, BA cannot be found.

2. How one entry is built

To get the (i, j)-th entry of AB, multiply the entries of the i-th row of A by the matching entries of the j-th column of B, then add. Take A = [[2, 2, −3], [5, 0, 2], [2, −1, 1]] and B = [[3, −1, 2], [4, 2, 5], [1, −3, −4]]. The (1, 2)-th entry is 2(−1) + 2·2 + (−3)(−3) = 11. The (2, 3)-th entry is 5·2 + 0·5 + 2(−4) = 2.

Done in full, AB = [[11, 11, 26], [17, −11, 2], [3, −7, −5]]. Tap the cells of AB below and check each one.

In Example 2, A = [[1, 2], [3, 4], [0, 1]] (3 × 2), B = [[4, 3], [2, 1]] and C = [[1, 2], [2, 3]]. Then AB = [[8, 5], [20, 13], [2, 1]] and BC = [[10, 17], [4, 7]]. But B has 2 columns and A has 3 rows. They do not match, so BA cannot be found.

3. AB and BA are not the same

See it before you read the rule. Keep A fixed, change B, and set AB beside BA. Any entry that differs is in red.

In general AB ≠ BA: matrix multiplication is not commutative. For A = [[1, 6], [−3, 5]] and B = [[4, 0], [2, −1]], AB = [[16, −6], [−2, −5]] but BA = [[4, 24], [5, 7]].

Sometimes, though, AB = BA = I. For A = [[3, −4, 2], [−2, 1, 0], [−1, −1, 1]] and B = [[1, 2, −2], [2, 5, −4], [3, 7, −5]], AB = BA = I₃.

4. Associativity, rows and columns, the identity

Multiplication is associative: (AB)C = A(BC). In Example 3, A = [[1, −1], [0, 2]], B = [[1, 3, 0], [2, 0, 1]] and C = [[2], [3], [4]]. Then AB = [[−1, 3, −1], [4, 0, 2]] and (AB)C = [[3], [16]]. The other way, BC = [[11], [8]] and A(BC) = [[3], [16]]. Both routes give the same answer.

A row matrix (1 × n) times a column matrix (n × 1) is a 1 × 1 matrix. A column times a row is an n × n matrix. For example [3, 2, −1] × [[4], [3], [8]] = [10], and [[3], [9], [0]] × [1, 2, 5] = [[3, 6, 15], [9, 18, 45], [0, 0, 0]].

In an expression with several matrices, work through the orders step by step. For A (4 × 5), B (5 × 4) and C (4 × 2), Aᵗ is 5 × 4 and B is 5 × 4, so Aᵗ + B is 5 × 4. Then (Aᵗ + B)C has order 5 × 2. Pick example 4 in the calculator above to check it.

For the identity matrix I, AI = IA = A and I² = I. So if A = I₃, then A² + 2A = I + 2I = 3A.

5. Powers and polynomials of a matrix

A power is just multiplying again. Press × A to go from A to A², then A³. When you reach A², see what A² + 2A − 11I comes to.

To evaluate a polynomial in a matrix, f(A), write the constant term as a multiple of I. For A = [[3, 2], [5, −1]], A² − 5A + 6I = [[19, 4], [10, 11]] − [[15, 10], [25, −5]] + [[6, 0], [0, 6]] = [[10, −6], [−15, 22]]. The constant term 6 had to be written 6I.

For A = [[1, 2], [4, −3]], A² = [[9, −4], [−8, 17]], A³ = [[−7, 30], [60, −67]] and A² + 2A − 11I = 0. For A = [[1, 2], [3, −4]], A² = [[7, −6], [−9, 22]], A³ = [[−11, 38], [57, −106]] and A² + 3A − 10I = 0. Pick both in the calculator and check.

(A + B)² = A² + AB + BA + B². Since AB may differ from BA, it cannot be written A² + 2AB + B². Take A = [[1, −1], [−1, 1]] and B = [[−1, 1], [1, −1]]: then A + B = 0 and B² = A². So (A + B)² = 0 = A² − B².

6. The book's profit sum, and finding unknowns

In Example 5(b), the profit matrix is P = selling price − cost = [38, 22, 62] − [27, 14, 39] = [11, 8, 23]. The sales matrix is B = [[2, 5, 3], [6, 7, 5], [5, 3, 4]], with rows jug, mug, bucket and columns Dhaka, Chattogram, Barishal. Then PB = [185, 180, 165]. The total daily profit is 530 taka, and for June it is 30 × 530 = 15900 taka.

Given a product, you can find the unknowns. If [[x, 3], [−2, y]] × [[2], [5]] = [[0], [0]], then row times column gives 2x + 15 = 0 and −4 + 5y = 0. So x = −15/2 and y = 4/5.

7. Check yourself

Three questions in the exam's style. In the first, match the orders.

In the second, work along rows and columns.

In the last one, watch the signs.

The whole note on six cards.

Source: Higher Mathematics 1st Paper, Classes 11-12 (NCTB), Chapter 1 “Matrices and Determinants”, section 1.2.