Page 14 - Focus SPM KSSM 2021 Tingkatan 5 - Maths DLP

Mathematics Form 5 Chapter 2 Matrices

D Explaining the characteristics of 22
identity matrix
It is given that matrix P = 3 12 4 4 and matrix
1. Identity matrix is a square matrix with its –1 2 –7 6
diagonal elements from top left corner to bottom Q = 3 3 –5 4 . Express each of the following as a
right corner are 1 and all other elements are 0. single matrix.

1 0
4
and 0
0 .
For example, 3 0 1 3 1 0 1 0 4 (a) PI + IQ
(b) (P – Q)I
0
1
0
Solution Chapter 2
2. The product of matrix A with identity matrix I is (a) PI + IQ
matrix A itself, which is
43
43
Chapter 2
+
AI = IA = A = 3 12 4 1 0 4 3 1 0 –1 2 4
–7 6 0 1
0 1 3 –5
+
20 = 3 12 4 4 3 –1 2 4
–7 6
3 –5
Write identity matrices for the following orders. 11 6
(a) 4 × 4 = 3 –4 1 4
(b) 1 × 1
(b) (P – Q)I
Solution
–
= 13 12 4 4 3 –1 2 423 1 0 4
1 0 0 0
0 1
3 –5
–7 6
3 0 0 1 04
0 1 0 0
(a) = 3 13 2 43 1 0 4
0 0 0 1 –10 11 0 1
= 3 13 2 4
(b) [1] –10 11
Try Questions 15 – 17 in Try This! 2.2
21
Determine whether the following matrices are identity
matrices for 3 3 5 4 . E Explaining the meaning of inverse
matrix and hence determining the
4 1
inverse matrix for a 2 × 2 matrix
(a) 3 1 0 4 (b) 3 0 1 4 1. When two matrices of order 2 × 2, A and B
0 1
1 0
Solution are multiplied, if AB = BA = I, then B is the
inverse matrix of A and A is the inverse matrix
43
(a) 3 3 5 1 0 4 3 3 + 0 0 + 5 4 of B.
=
4 1 0 1
4 + 0 0 + 1
−1
= 3 3 5 4 2. Inverse matrix of A is written as A .
Then, AA = A A = I.
−1
−1
4 1
Hence, 3 1 0 4 is an identity matrix. 3. When matrix A = 3 a b 4 , the inverse matrix of
0 1
c d
A, A can be derived by the formula below:
−1
43
(b) 3 3 5 0 1 4 3 0 + 5 3 + 0 4 A = 1 3 d –b 4
=
0 + 1 4 + 0
4 1 1 0
−1
= 3 5 3 4 where ad – bc ≠ 0. ad – bc –c a
1 4
Hence, 3 0 1 4 is not an identity matrix. 4. ad – bc is known as determinant of matrix A,
1 0
|A|, where ad – bc ≠ 0.
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