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3.1 Overview
3.1.1 A matrix is an ordered rectangular array of numbers (or functions). For example,
A =
x x x
The numbers (or functions) are called the elements or the entries of the matrix.
The horizontal lines of elements are said to constitute rows of the matrix and the vertical lines of elements are said to constitute columns of the matrix.
3.1.2 Order of a Matrix
A matrix having m rows and n columns is called a matrix of order m × n or simply m × n matrix (read as an m by n matrix).
In the above example, we have A as a matrix of order 3 × 3 i.e., 3 × 3 matrix.
In general, an m × n matrix has the following rectangular array :
A = [ aij ] m × n =
11 12 13 1 21 22 23 2
(^1 2 3) ×
n n
m m m mn (^) m n
a a a a a a a a
a a a a
1 ≤ i ≤ m , 1≤ j ≤ n i , j ∈ N.
The element, aij is an element lying in the i th^ row and j th^ column and is known as the ( i , j )th^ element of A. The number of elements in an m × n matrix will be equal to mn.
3.1.3 Types of Matrices
(i) A matrix is said to be a row matrix if it has only one row.
Chapter 3
Matrices
MATRICES 43
(ii) A matrix is said to be a column matrix if it has only one column. (iii) A matrix in which the number of rows are equal to the number of columns, is said to be a square matrix. Thus, an m × n matrix is said to be a square matrix if m = n and is known as a square matrix of order ‘ n ’. (iv) A square matrix B = [ bij ] n×n is said to be a diagonal matrix if its all non diagonal elements are zero, that is a matrix B = [ bij ] n×n is said to be a diagonal matrix if bij = 0, when i ≠ j. (v) A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal, that is, a square matrix B = [ bij ] n×n is said to be a scalar matrix if bij = 0, when i ≠ j bij = k , when i = j , for some constant k. (vi) A square matrix in which elements in the diagonal are all 1 and rest are all zeroes is called an identity matrix. In other words, the square matrix A = [ aij ] n×n is an identity matrix, if aij = 1, when i = j and aij = 0, when i ≠ j. (vii) A matrix is said to be zero matrix or null matrix if all its elements are zeroes. We denote zero matrix by O. (ix) Two matrices A = [ aij ] and B = [ bij ] are said to be equal if (a) they are of the same order, and (b) each element of A is equal to the corresponding element of B, that is, aij = bij for all i and j.
3.1.4 Additon of Matrices
Two matrices can be added if they are of the same order.
3.1.5 Multiplication of Matrix by a Scalar
If A = [ aij ] (^) m×n is a matrix and k is a scalar, then k A is another matrix which is obtained by multiplying each element of A by a scalar k , i.e. k A = [ kaij ] m × n
3.1.6 Negative of a Matrix
The negative of a matrix A is denoted by –A. We define –A = (–1)A.
3.1.7 Multiplication of Matrices
The multiplication of two matrices A and B is defined if the number of columns of A is equal to the number of rows of B.
MATRICES 45
(ii) A square matrix A = [ aij ] is said to be skew symmetric matrix if AT^ = –A, that is aji = – aij for all possible values of i and j. Note : Diagonal elements of a skew symmetric matrix are zero. (iii) Theorem 1: For any square matrix A with real number entries, A + AT^ is a symmetric matrix and A – AT^ is a skew symmetric matrix. (iv) Theorem 2: Any square matrix A can be expressed as the sum of a symmetric matrix and a skew symmetric matrix, that is
(A + A )^ T^ (A A )T A = + 2 2
3.1.10 Invertible Matrices
(i) If A is a square matrix of order m × m , and if there exists another square matrix B of the same order m × m , such that AB = BA = I m , then, A is said to be invertible matrix and B is called the inverse matrix of A and it is denoted by A–1. Note :
- A rectangular matrix does not possess its inverse, since for the products BA and AB to be defined and to be equal, it is necessary that matrices A and B should be square matrices of the same order.
- If B is the inverse of A, then A is also the inverse of B. (ii) Theorem 3 (Uniqueness of inverse) Inverse of a square matrix, if it exists, is unique. (iii) Theorem 4 : If A and B are invertible matrices of same order, then (AB)–1^ = B–1A–1.
3.1.11 Inverse of a Matrix using Elementary Row or Column Operations
To find A–1^ using elementary row operations, write A = IA and apply a sequence of row operations on (A = IA) till we get, I = BA. The matrix B will be the inverse of A. Similarly, if we wish to find A–^1 using column operations, then, write A = AI and apply a sequence of column operations on A = AI till we get, I = AB.
Note : In case, after applying one or more elementary row (or column) operations on A = IA (or A = AI), if we obtain all zeros in one or more rows of the matrix A on L.H.S., then A–1^ does not exist.
46 MATHEMATICS
3.2 Solved Examples
Short Answer (S.A.)
Example 1 Construct a matrix A = [ aij ]2×2 whose elements aij are given by
aij = e^2 ix sin jx.
Solution For i = 1, j = 1, a 1 1 = e^2 x^ sin x
For i = 1, j = 2, a 1 2 = e^2 x^ sin 2 x For i = 2, j = 1, a 2 1 = e^4 x^ sin x For i = 2, j = 2, a 2 2 = e^4 x^ sin 2 x
Thus A =
2 2 4 4
sin sin 2 sin sin 2
x x x x
e x e x e x e x
Example 2 If A =
, B =
, C =
, D =
, then
which of the sums A + B, B + C, C + D and B + D is defined?
Solution Only B + D is defined since matrices of the same order can only be added.
Example 3 Show that a matrix which is both symmetric and skew symmetric is a zero matrix.
Solution Let A = [ aij ] be a matrix which is both symmetric and skew symmetric.
Since A is a skew symmetric matrix, so A′ = –A.
Thus for all i and j , we have aij = – aji. (1)
Again, since A is a symmetric matrix, so A′ = A.
Thus, for all i and j , we have
aji = aij (2)
Therefore, from (1) and (2), we get
aij = – aij for all i and j
or 2 aij = 0,
i.e., aij = 0 for all i and j. Hence A is a zero matrix.
48 MATHEMATICS
Hence
A + A
and
A – A
Therefore,
A A A A 11 3 3 + 3 0 7 7 3 5 A
−^ −^ −
^ −
+ ′^ − ′
− − ^ −
Example 7 If A =
, then show that A satisfies the equation
A^3 –4A^2 –3A+11I = O.
Solution A^2 = A × A =
2 0 1 × 2 0 1
MATRICES 49
and A^3 = A^2 × A =
1 4 1 × 2 0 1
+^ +^ +^ +^ −^ +
Now A^3 – 4A^2 – 3A + 11(I)
−^ −^ +^ −^ −^ +^ −^ −^ +
MATRICES 51
Objective Type Questions
Choose the correct answer from the given four options in Examples 9 to 12.
Example 9 If A and B are square matrices of the same order, then
(A + B) (A – B) is equal to
(A) A^2 – B^2 (B) A^2 – BA – AB – B^2 (C) A^2 – B^2 + BA – AB (D) A^2 – BA + B^2 + AB
Solution (C) is correct answer. (A + B) (A – B) = A (A – B) + B (A – B) = A^2 – AB + BA – B^2
Example 10 If A =
(^) − and B =
, then
(A) only AB is defined (B) only BA is defined (C) AB and BA both are defined (D) AB and BA both are not defined.
Solution (C) is correct answer. Let A = [ aij ]2×3 B = [ bij ]3×2.^ Both AB and BA are defined.
Example 11 The matrix A =
is a
(A) scalar matrix (B) diagonal matrix (C) unit matrix (D) square matrix
Solution (D) is correct answer.
Example 12 If A and B are symmetric matrices of the same order, then (AB′ –BA′) is a
(A) Skew symmetric matrix (B) Null matrix (C) Symmetric matrix (D) None of these
Solution (A) is correct answer since
(AB′ –BA′)′ = (AB′)′ – (BA′)′
52 MATHEMATICS
= (BA′ – AB′)
= – (AB′ –BA′)
Fill in the blanks in each of the Examples 13 to 15:
Example 13 If A and B are two skew symmetric matrices of same order, then AB is symmetric matrix if ________.
Solution AB = BA.
Example 14 If A and B are matrices of same order, then (3A –2B)′ is equal to ________.
Solution 3A′ –2B′.
Example 15 Addition of matrices is defined if order of the matrices is ________
Solution Same.
State whether the statements in each of the Examples 16 to 19 is true or false:
Example 16 If two matrices A and B are of the same order, then 2A + B = B + 2A.
Solution True
Example 17 Matrix subtraction is associative
Solution False
Example 18 For the non singular matrix A, (A′)–1^ = (A–1)′.
Solution True
Example 19 AB = AC ⇒ B = C for any three matrices of same order.
Solution False
3.3 EXERCISE
Short Answer (S.A.)
1. If a matrix has 28 elements, what are the possible orders it can have? What if it has 13 elements? 2. In the matrix A =
2
a x x y
, write :
54 MATHEMATICS
10. Find the value of x if
[ 1 x^1 ]
x
= O.
11. Show that A =
satisfies the equation AA^2 – 3A – 7I = O and hence find A–1.
12. Find the matrix A satisfying the matrix equation:
2 1 3 2 1 0 A = 3 2 5 3 0 1
13. Find A, if
A =
14. If A =
and B =
, then verify (BA)^2 ≠^ B^2 A^2
15. If possible, find BA and AB, where
A =
, B =
16. Show by an example that for A ≠ O, B ≠ O, AB = O. 17. Given A =
and B =
. Is (AB)′ = B′A′? 18. Solve for x and y :
MATRICES 55
O
x y
+^ +^ =
19. If X and Y are 2 × 2 matrices, then solve the following matrix equations for X and Y
2X + 3Y =
, 3X + 2Y =
20. If A = (^) [ 3 5 ] , B = (^) [ 7 3 ] , then find a non-zero matrix C such that AC = BC. 21. Give an example of matrices A, B and C such that AB = AC, where A is non- zero matrix, but B ≠ C. 22. If A =
, B =
and C =
, verify :
(i) (AB) C = A (BC) (ii) A (B + C) = AB + AC.
23. If P =
x y z
and Q =
a b c
, prove that
PQ =
xa yb zc
= QP..
24. If : (^) [ 2 1 3 ]
−^ −
= A, find A.A.
25. If A = (^) [ 2 1 ] , B =
and C =
, verify that A (B + C) = (AB + AC).
MATRICES 57
(c) ( a + b )B = a B + b B (d) a (C–A) = a C – a A (e) (AT)T^ = A (f) ( b A)T^ = b AT (g) (AB)T^ = BT^ AT (h) (A –B)C = AC – BC (i) (A – B)T^ = AT^ – BT
33. If A =
cosθ sinθ
, then show that AA^2 =
cos2θ sin2θ
34. If A =
x x
,^ B =^
and^ x^2 = –1, then show that (A + B)^2 = AA^2 + B^2.
35. Verify that A^2 = I when A =
36. Prove by Mathematical Induction that (A′) n^ = (A n )′, where n ∈ N for any square matrix A. 37. Find inverse, by elementary row operations (if possible), of the following matrices
(i)
(ii)
38. If
xy z x y
+ + =^
^ w , then find values of^ x ,^ y ,^ z^ and^ w.
39. If A =
and B =
, find a matrix C such that 3A + 5B + 2C is a null matrix.
58 MATHEMATICS
40. If A =
(^) − , then find AA^2 – 5A – 14I. Hence, obtain A^3.
41. Find the values of a , b , c and d , if
a b c d
a d
a b c d
42. Find the matrix A such that
2 1 1 0 3 4
A =
−^ −^ −
43. If A =
, find AA^2 + 2A + 7I.
44. If A =
cos α sinα sinα cosα
, and A – 1^ = A′ , find value of α.
45. If the matrix
a b c
is a skew symmetric matrix, find the values of a , b and c.
46. If P ( x ) =
cos sin sin cos
x x x x
, then show that
P ( x ). P ( y ) = P ( x + y ) = P ( y ). P ( x ).
47. If A is square matrix such that A^2 = A, show that (I + A)^3 = 7A + I. 48. If A, B are square matrices of same order and B is a skew-symmetric matrix, show that A′BA is skew symmetric.
Long Answer (L.A.)
49. If AB = BA for any two sqaure matrices, prove by mathematical induction that (AB) n^ = A n^ B n.
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56. If A =
1 1
1 1
sin ( ) tan 1 sin cot ( )
x^ x
x (^) x
− −
− −
^
, B =
1 1
1 1
cos ( ) tan 1 sin tan ( )
x^ x
x (^) x
− −
− −
^
, then
A – B is equal to
(A) I (B) O (C) 2I (D)
1 I
57. If A and B are two matrices of the order 3 × m and 3 × n , respectively, and m = n , then the order of matrix (5A – 2B) is (A) m × 3 (B) 3 × 3 (C) m × n (D) 3 × n 58. If A =
, then AA^2 is equal to
(A)
(B)
(C)
(D)
59. If matrix A = [ aij ]2 × 2, where aij = 1 if i ≠ j
= 0 if i = j then A^2 is equal to (A) I (B) A (C) 0 (D) None of these
60. The matrix
is a
(A) identity matrix (B) symmetric matrix (C) skew symmetric matrix (D) none of these
MATRICES 61
61. The matrix
is a
(A) diagonal matrix (B) symmetric matrix (C) skew symmetric matrix (D) scalar matrix
62. If A is matrix of order m × n and B is a matrix such that AB′ and B′A are both defined, then order of matrix B is (A) m × m (B) n × n (C) n × m (D) m × n 63. If A and B are matrices of same order, then (AB′–BA′) is a
(A) skew symmetric matrix (B) null matrix (C) symmetric matrix (D) unit matrix
64. If A is a square matrix such that A^2 = I, then (A–I)^3 + (A + I)^3 –7A is equal to
(A) A (B) I – A (C) I + A (D) 3A
65. For any two matrices A and B, we have
(A) AB = BA (B) AB ≠ BA (C) AB = O (D) None of the above
66. On using elementary column operations C 2 → C 2 – 2C 1 in the following matrix equation
1 3 2 4
, we have :
(A)
(B)
