Higher Mathematics 1st Paper · Classes 11-12 · Chapter 1 · 1.7-1.8
Singular and Non-singular Matrices and the Idea of an Inverse.
In [[6, 4], [3, 2]] the second row is half the first, so it adds nothing new. A matrix like that cannot be inverted. How do you know before you start working? One number tells you: the determinant.
1. Singular or non-singular
A square matrix A is singular if |A| = det(A) = 0 and non-singular if |A| ≠ 0. For example A = [[7, 4], [14, 8]] has |A| = 56 − 56 = 0, so it is singular. B = [[5, 6], [10, 13]] has |B| = 65 − 60 = 5, so it is non-singular.
Example 3: A = [[−8, 5], [7, −4]] has |A| = 32 − 35 = −3 ≠ 0, so A is non-singular and invertible. Example 4: B = [[6, 4], [3, 2]] has |B| = 12 − 12 = 0, so B is singular and B⁻¹ cannot be found.
Exercise 1(B) 10: [[6, 2], [9, 3]], [[2, 5], [10, 25]] and [[28, 29, 30], [31, 33, 35], [34, 37, 40]] have determinant 0, so they are not invertible. [[4, 4, 0], [3, 7, 6], [−3, 11, 15]] has determinant −96 ≠ 0, so it is invertible. Now sort them yourself: work out the determinant, then drop the chip in its box.
Singular or non-singular?
2. For which value is it singular?
When a matrix has an unknown and is said to be singular, set the determinant to 0 and solve the equation. Exercise 1(B) 11: if [[x + 5, 5], [3, x − 9]] is singular then (x + 5)(x − 9) − 15 = x² − 4x − 60 = 0, so x = 10 or −6.
Recall the picture from topic 1.3: the two columns span a parallelogram, and its area is the determinant. Now slide the unknown and see what happens at x = 10 or −6.
When does the shape go flat?
More second-order cases: [[m − 2, 6], [2, m − 3]] singular gives m² − 5m − 6 = (m − 6)(m + 1) = 0, m = 6, −1. [[k − 3, −2], [−2, k − 2]] not invertible gives k² − 5k + 2 = 0, k = (5 ± √17)/2. [[2 − x, 13], [5, 10 − x]] singular gives x² − 12x − 45 = 0, x = 15, −3.
Third order follows the same road. [[1, 2, a], [4, 5, 6], [7, 8, 9]] singular gives 9 − 3a = 0, a = 3. [[1, λ, 3], [2, 1, λ], [5, 4, 7]] singular gives 5λ² − 18λ + 16 = (λ − 2)(5λ − 8) = 0, λ = 2 or 8/5. All six matrices are below; pick one and try a value of your own to see the determinant.
Make the determinant zero
The matrices come from the book's examples and exercises. The working is the machine's, not a measurement of you. Nothing goes anywhere; the notebook is yours.
3. The inverse matrix
If the product of two matrices is the identity matrix, each is called the inverse of the other. For a square matrix A, if a matrix B of the same order has AB = BA = I, then B is the inverse matrix of A, written A⁻¹.
Not every matrix has one: only a non-singular matrix has an inverse. A singular matrix has none.
Check the definition by hand. For A = [[−5/2, 2], [3/2, −1]] and B = [[2, 4], [3, 5]], AB = [[−5 + 6, −10 + 10], [3 − 3, 6 − 5]] = [[1, 0], [0, 1]] = I, and BA = I too. So A = B⁻¹ and B = A⁻¹.
Task 1: B = [[2, 3], [4, 6]] has determinant 0. C = [[1, 4, 6], [2, 6, 4], [1, 4, 6]] has rows 1 and 3 equal, determinant 0. Neither has an inverse. A = [[−1, 2, −3], [2, 1, 0], [4, −2, 5]] has determinant −1 ≠ 0, so it has an inverse.
4. The inverse of order two
The inverse of a second-order square matrix [[a, b], [c, d]] is (1/(ad − bc)) [[d, −b], [−c, a]]. Swap the diagonal, flip the other two, divide by the determinant: the two diagonal entries trade places, the other two change sign.
Exercise 1(B) 12: [[3, 1], [7, 5]] has determinant 8 and inverse [[5/8, −1/8], [−7/8, 3/8]]. [[3, 7], [2, 6]] has determinant 4 and inverse [[3/2, −7/4], [−1/2, 3/4]].
The same formula works when entries are complex. [[i, i], [2i, i]] (i² = −1) has determinant i·i − i·2i = −i² = 1, so its inverse is [[i, −i], [−2i, i]]. And [[4, 3], [3, 2]] has determinant −1 and inverse [[−2, 3], [3, −4]]. Change the four boxes below and build your own; the machine also checks A·A⁻¹.
Build the inverse
The matrices come from the book's exercises and MCQs. The working is the machine's, not a measurement of you. Nothing goes anywhere; the notebook is yours.
The method for inverting larger matrices, and the rules for inverses, come in the next note.
5. Check yourself
The first question is in the exam's style.
Question 1
A = [[2, 3, 4], [4, 6, 8], [3, 1, 2]] is a:
i. a square matrix
ii. a singular matrix
iii. a symmetric matrix
In the second, write the determinant first.
Question 2
Which of the following is a singular matrix?
In the last one, mind the signs.
Question 3
For which value of k is the matrix [[k + 1, 3], [3, k − 1]] not invertible?
The whole note on seven cards.
Seven cards
Source: Higher Mathematics 1st Paper, Classes 11-12 (NCTB), Chapter 1 “Matrices and Determinants”, section 1.7-1.8.