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Practice each question, then open the answer or worked solution.
Find the value of a, b, c and d, if
Given,
Now,
3a = a + 4 ⇒ a = 2
3b = 6 + a + b
3b – b = 8
⇒ b = 4
And,
3d = 3 + 2d
⇒ d = 3
And,
3c = c + d – 1
2c = 3 – 1 = 2
⇒ c = 1
Hence,
a = 2, b = 4, c = 1 and d = 3
Find the matrix A such that
Let,
Now, by equality of matrices, we get
a = 1, b = -2, c = -5
And,
2a – d = -1 ⇒ d = 2a + 1 = 3;
2b – e = -8 ⇒ e = 2(-2) + 8 = 4
2c – f = -10 ⇒ f = 2c + 10 = 0
Thus,
If A = find A2 + 2A + 7I.
Given,
If A = and A-1 = A’, find the value of a.
By using the equality of matrices, we get
cos2 α + sin2 α = 1, which is true for all real values of α.
If the matrix is a skew symmetric matrix, find the values of a, b and c.
If P (x) = then show that
P (x) . P (y) = P (x + y) = P (y) . P (x).
Given,
If A is square matrix such that A2 = A, show that (I + A)3 = 7A + I.
We know that,
A . I = I . A
So, A and I are commutative.
Thus, we can expand (I + A)3 like real numbers expansion.
So, (I + A)3 = I3 + 3I2A + 3IA2 + A3
= I + 3IA + 3A2 + AA2 (As In = I, n ∈ N)
= I + 3A + 3A + AA
= I + 3A + 3A + A2 = I + 3A + 3A + A = I + 7A
If A, B are square matrices of same order and B is a skew-symmetric matrix,
show that A¢BA is skew symmetric.
Given, A and B are square matrices such that B is a skew-symmetric matrix
So, B¢ = -B
Now, we have to prove that A¢BA is a skew-symmetric matrix.
(A¢BA) ¢ = A¢B¢ (A¢)¢ [Since, (AB) ¢ = B¢A¢]
= A¢ (-B)A
= -A¢BA
Hence, A¢BA is a skew-symmetric matrix.
Long Answer (L.A)
If AB = BA for any two square matrices, prove by mathematical induction that (AB)n = An Bn.
Let P(n) : (AB)n = AnBn
So, P(1) : (AB)1 = A1B1
AB = AB
So, P(1) is true.
Let P(n) is true for some k ∈ N
Now,
Hence, P(1) is true and whenever P(k) is true P(k + 1) is true.
Thus, P(n) is true for all n ∈ N.
Find x, y, z if A = satisfies A¢ = A-1.
Matrix A is such that A¢ = A-1
AA¢ = I
