Class 12 Mathematics · 103 questions · 0 with answers
Construct a matrix A = [a ij] 2×2 whose elements a ij are given by aij = e2ix sin jx . Solution For i = 1, j = 1, a 11 = e2x sin x For i = 1, j = 2, a 12 = e2x sin 2x For i = 2, j = 1, a 21 = e4x sin x For i = 2, j = 2, a 22 = e4x sin 2x e 2 x sin x e 2 x sin 2 x Thus A = 4x 4x e sin x e sin 2 x 2 3 1 3 2 1 4 6 8
If A = ,B= , C= , D= , then 1 2 4 3 1 2 5 7 9 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.
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 2aij = 0, i.e., aij = 0 for all i and j. Hence A is a zero matrix. MATRICES 47 1 2 x
If [ 2 x 3] = O , find the value of x. –3 0 8 Solution We have 1 2 x x [ 2 x 3] –3 0 8 = O ⇒ [ 2 x − 9 4 x ] 8 = [ 0] or 2 x 2 − 9 x +32 x = [ 0] ⇒ 2 x 2 + 23x = 0 −23 or x(2 x + 23) = 0 ⇒ x = 0, x =
If A is 3 × 3 invertible matrix, then show that for any scalar k (non-zero),
Express the matrix A as the sum of a symmetric and a skew symmetric matrix, where 2 4 −6 A = 7 3 5 . 1 −2 4 Solution We have 2 4 −6 2 7 1 A = 7 3 5 , then A′ = 4 3 −2 1 −2 4 −6 5 4 48 MATHEMATICS 11 −5 2 2 2 4 11 −5 11 3 A + A′ 1 11 6 3 = 2 3 2
If A = 2 0 −1 , then show that A satisfies the equation 1 2 3 A3–4A2–3A+11I = O. 1 3 2 1 3 2 Solution A2 = A × A = 2 0 −1 × 2 0 −1 1 2 3 1 2 3 MATRICES 49 1 + 6 + 2 3+0+ 4 2 − 3 + 6 2 + 0 − 1 6+0−2 4 + 0 − 3 = 1 + 4 + 3 3+0+6 2 − 2 + 9 9 7 5 = 1 4 1 8 9 9 9 7 5 1 3 2 and A3 = A2 × A = 1 4 1 × 2 0 −1 8 9 9 1 2 3 9 + 14 + 5 27 + 0 + 10 18 − 7 + 15 1+ 8 +1 3+ 0+ 2 2 − 4 + 3 = 8 + 18 + 9 24 + 0 + 18 16 − 9 + 27 28 37 26 = 10 5 1 35 42 34 Now A3 – 4A2 – 3A + 11(I) 28 37 26 9 7 5 1 3 2 1 0 0 10 5 1 – 4 1 4 1 –3 2 0 −1 +11 0 1 0 = 35 42 34 8 9 9 1 2 3 0 0 1 28 − 36 − 3 + 11 37 − 28 − 9 + 0 26 − 20 − 6 + 0 10 − 4 − 6 + 0 5 − 16 + 0 + 11 1 − 4 + 3 + 0 = 35 − 32 − 3 + 0 42 − 36 − 6 + 0 34 − 36 − 9 + 11 50 MATHEMATICS 0 0 0 = 0 0 0 = O 0 0 0 2 3
Let A = . Then show that A2 – 4A + 7I = O. –1 2 Using this result calculate A5 also. 2 2 3 2 3 1 12 Solution We have A = = , −1 2 −1 2 − 4 1 −8 −12 7 0 − 4A= and 7 I = . 4 −8 0 7 1 − 8 + 7 12 −12 + 0 0 0 Therefore, A2 – 4A + 7I = = =O −4 + 4 + 0 1 − 8 + 7 0 0 ⇒ A2 = 4A – 7I Thus A3 = A.A2 = A (4A – 7I) = 4 (4A – 7I) – 7A = 16A – 28I – 7A = 9A – 28I and so A5 = A3A2 = (9A – 28I) (4A – 7I) = 36A2 – 63A – 112A + 196I = 36 (4A – 7I) – 175A + 196I = – 31A – 56I 2 3 1 0 = − 31 − 56 −1 2 0 1 −118 −93 = 31 −118 MATRICES 51
If A and B are square matrices of the same order, then (A + B) (A – B) is equal to
If A = and B = 4 −2 , then −4 5 1 1 5
The matrix A = 0 5 0 is a 5 0 0
If A and B are symmetric matrices of the same order, then (AB′ –BA′)
If A and B are two skew symmetric matrices of same order, then AB is symmetric matrix if ________. Solution AB = BA.
If A and B are matrices of same order, then (3A –2B)′ is equal to ________. Solution 3A′ –2B′.
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:
If two matrices A and B are of the same order, then 2A + B = B + 2A. Solution True
Matrix subtraction is associative Solution False
For the non singular matrix A, (A′)–1 = (A–1)′. Solution True
AB = AC ⇒ B = C for any three matrices of same order. Solution False
–1 kA is invertible and (kA)–1 = A Solution We have 1 –1 1 (kA) A = k . (A. A–1) = 1 (I) = I k k 1 –1 1 –1 Hence (kA) is inverse of A or (kA)–1 = A k k
−5 3 8 −5 3 Hence = 4 2 2 −3 −7 0 2 2 0 −3 −7 3 7 A – A′ 1 3 0 7 = 2 0 2 2 7 −7 0 7 and = −7 0 2 2 Therefore, 11 −5 −3 −7 2 2 2 2 2 2 4 −6 A + A′ A − A′ 7 = = 7 3 5 = A 11 3 3 + 3 + 0 2 2 2 2 2 2 1 −2 4 . −5 3 4 7 −7 0 2 2 2 2 1 3 2
If a matrix has 28 elements, what are the possible orders it can have? What if it has 13 elements? a 1 x 2 3 x2 − y In the matrix A = −2
, write : 0 5 5 MATRICES 53
Construct a2 × 2 matrix where (i − 2 j ) 2
Construct a 3 × 2 matrix whose elements are given by aij = ei.xsinjx
Find values of a and b if A = B, where a + 4 3b 2a + 2 b2 + 2 A= , B= 8 8 −6 b 2 − 5b 3 1
If possible, find the sum of the matrices A and B, where A = 3 , 2 x y z and B = a b 6 3 1 −1 2 1 −1 X= and Y = 4
If , find 5 −2 −3 7 2
Find non-zero values of x satisfying the matrix equation: 2 x 2 8 5 x ( x 2 + 8) 24 x + 2 4 4 x = 2 3 x (10) 6 x . 0 1 0 −1
If A = and B = , show that (A + B) (A – B) ≠ A2 – B2. 1 1 1 0 54 MATHEMATICS
Find the value of x if 1 3 2 1 2 [1 x 1] 2 5 1 = O. 15 3 2 x 5 3
Show that A = satisfies the equation A2 – 3A – 7I = O and hence −1 −2 find A–1.
Find the matrix A satisfying the matrix equation: 2 1 −3 2 1 0 3 2 A 5 −3 = 0 1 4 −4 8 4 1
Find A, if A = −1 2 1 3 −3 6 3 3 −4 2 1 2
If A = 1 1 and B = , then verify (BA)2 ≠ B2A2 2 0 1 2 4
If possible, find BA and AB, where 4 1 2 1 2 A= , B = 2 3 . 1 2 4 1 2
Show by an example that for A ≠ O, B ≠ O, AB = O. 1 4 2 4 0
Given A = and B = 2 8 . Is (AB)′ = B′A′? 3 9 6 1 3
Solve for x and y: MATRICES 55 2 3 −8 x + y + = O. 1 5 −11
If X and Y are 2 × 2 matrices, then solve the following matrix equations for X and Y 2 3 −2 2 2X + 3Y = , 3X + 2Y = . 4 0 1 −5
If A = [ 3 5] , B = [ 7 3] , then find a non-zero matrix C such that AC = BC.
Give an example of matrices A, B and C such that AB = AC, where A is non- zero matrix, but B ≠ C. 1 2 2 3 1 0
If A = , B= and C = , verify : −2 1 3 −4 −1 0
If P = and Q = 0 b 0 , prove that 0 0 z 0 0 c xa 0 0 yb 0 = QP.. PQ = 0 0 0 zc −1 0 −1 1 −1 1 0 0
If : [ 2 1 3] = A, find A. 0 1 1 −1 5 3 4 −1 2 1
If A = [ 2 1] , B = 8 7 6 and C = 1 0 2 , verify that A (B + C) = (AB + AC). 56 MATHEMATICS 1 0 −1
If A = 2 1 3 , then verify that A2 + A = A (A + I), where I is 3 × 3 unit 0 1 1 matrix. 4 0 0 −1 2
If A = and B = 1 3 , then verify that : 4 3 −4 2 6
If A = , B = 6 4 , then verify that : 5 6 7 3
Show that A′A and AA′ are both symmetric matrices for any matrix A.
Let A and B be square matrices of the order 3 × 3. Is (AB)2 = A2 B2 ? Give reasons.
Show that if A and B are square matrices such that AB = BA, then (A + B)2 = A2 + 2AB + B2. 1 2 4 0 2 0
Let A = , B= , C = and a = 4, b = –2. −1 3 1 5 1 −2 Show that:
If A = , then show that A2 = . – sinθ cosθ – sin2θ cos2θ 0 − x 0 1
If A = , B= and x2 = –1, then show that (A + B)2 = A2 + B2. x 0 1 0 0 1 −1
Verify that A2 = I when A = 4 −3 4 . 3 −3 4
Prove by Mathematical Induction that (A′)n = (An)′, where n ∈ N for any square matrix A.
Find inverse, by elementary row operations (if possible), of the following matrices 1 3 1 −3
If = , then find values of x, y, z and w. z + 6 x + y 0 6 1 5 9 1
If A = and B = , find a matrix C such that 3A + 5B + 2C is a null 7 12 7 8 matrix. 58 MATHEMATICS 3 −5
If A = , then find A2 – 5A – 14I. Hence, obtain A3. −4 2
Find the values of a, b, c and d, if a b a 6 4 a + b 3 = + 3 . c d −1 2 d c + d
Find the matrix A such that 2 −1 −1 −8 −10 1 0 1 −2 −5 A = . −3 4 9 22 15 1 2
If A = , find A2 + 2A + 7I. 4 1 cos α sinα
If A = , and A – 1 = A′ , find value of α. −sinα cosα 0 a 3
If the matrix 2 b −1 is a skew symmetric matrix, find the values of a, b and c. c 1 0 cos x sinx
If P (x) = , then show that −sinx cosx P (x) . P (y) = P (x + y) = P (y) . P (x).
If A is square matrix such that A2 = A, show that (I + A)3 = 7A + I.
If A, B are square matrices of same order and B is a skew-symmetric matrix, show that A′BA is skew symmetric.
If AB = BA for any two sqaure matrices, prove by mathematical induction that (AB)n = An Bn. MATRICES 59 0 2 y z
Find x, y, z if A = x y − z satisfies A′ = A–1. x − y z
If possible, using elementary row transformations, find the inverse of the following matrices 2 −1 3 2 3 −3 2 0 −1
Express the matrix 1 −1 2 as the sum of a symmetric and a skew symmetric 4 1 2 matrix.
to 67. 0 0 4 53. The matrix P = 0 4 0 is a 4 0 0
Total number of possible matrices of order 3 × 3 with each entry 2 or 0 is
If = x + 6 , then the value of x + y is 5x − 7 4 x y
If A = −1 x , B = −1 x , then sin cot ( x) sin − tan ( x) −1 −1 A – B is equal to
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
If A = , then A2 is equal to 1 0 0 1 1 0
If matrix A = [aij]2 × 2, where aij = 1 if i ≠ j = 0 if i = j then A2 is equal to
The matrix 0 2 0 is a 0 0 4
The matrix 5 −8 −12 0
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
If A and B are matrices of same order, then (AB′–BA′) is a
If A is a square matrix such that A2 = I, then (A–I)3 + (A + I)3 –7A is equal to
For any two matrices A and B, we have
On using elementary column operations C2 → C2 – 2C1 in the following matrix equation 1 −3 1 −1 3 1 2 4 = 0 1 2 4 , we have : 1 −5 1 −1 3 −5
On using elementary row operation R1 → R1 – 3R2 in the following matrix equation: 4 2 1 2 2 0 3 3 = 0 3 1 1 , we have : −5 −7 1 −7 2 0
_________ matrix is both symmetric and skew symmetric matrix.
Sum of two skew symmetric matrices is always _________ matrix.
The negative of a matrix is obtained by multiplying it by _________.
The product of any matrix by the scalar _________ is the null matrix.
A matrix which is not a square matrix is called a _________ matrix.
Matrix multiplication is _________ over addition.
If A is a symmetric matrix, then A3 is a _________ matrix.
If A is a skew symmetric matrix, then A2 is a _________. MATRICES 63
If A and B are square matrices of the same order, then
If A is skew symmetric, then kA is a _________. (k is any scalar)
If A and B are symmetric matrices, then
If A is symmetric matrix, then B′AB is _________.
If A and B are symmetric matrices of same order, then AB is symmetric if and only if _________.
In applying one or more row operations while finding A–1 by elementary row operations, we obtain all zeros in one or more, then A–1 _________. State Exercises 82 to 101 which of the following statements are True or False
A matrix denotes a number.
Matrices of any order can be added.
Two matrices are equal if they have same number of rows and same number of columns.
Matrices of different order can not be subtracted.
Matrix addition is associative as well as commutative.
Matrix multiplication is commutative.
A square matrix where every element is unity is called an identity matrix.
If A and B are two square matrices of the same order, then A + B = B + A.
If A and B are two matrices of the same order, then A – B = B – A.
If matrix AB = O, then A = O or B = O or both A and B are null matrices.
Transpose of a column matrix is a column matrix.
If A and B are two square matrices of the same order, then AB = BA.
If each of the three matrices of the same order are symmetric, then their sum is a symmetric matrix. 64 MATHEMATICS
If A and B are any two matrices of the same order, then (AB)′ = A′B′.
If (AB)′ = B′ A′, where A and B are not square matrices, then number of rows in A is equal to number of columns in B and number of columns in A is equal to number of rows in B.
If A, B and C are square matrices of same order, then AB = AC always implies that B = C.
AA′ is always a symmetric matrix for any matrix A. 2 3 2 3 −1 5 , then AB and BA are defined and equal.
If A = and B = 4 1 4 2 2 1
If A is skew symmetric matrix, then A2 is a symmetric matrix.
(AB)–1 = A–1. B–1, where A and B are invertible matrices satisfying commutative property with respect to multiplication.