Q21Written answer
Give an example of matrices A, B and C such that AB = AC, where A is non-zero
matrix, but B ≠ C.
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Worked solution
From (i) and (ii),
Hence, AB = AC but B ≠ C.
NCERT Exemplar · Class 12 · Maths
Chapter 3: Matrices
Practice each question, then open the answer or worked solution.
Questions
50 questionsQ22Written answer
If
(i) (AB) C = A (BC) (ii) A (B + C) = AB + AC.
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23.
Q23Written answer
If \( P = \begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix} \) and \( Q = \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix} \), prove that \( PQ = QP \).
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Solution coming soon.
Q24Written answer
If: = A, find A.
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Q25Written answer
If verify that
A (B + C) = (AB + AC).
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Q26Written answer
If then verify that A2 + A = A (A + I), where I is 3 × 3 unit matrix.
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Q27Written answer
If then verify that:
(i) (A¢)¢ = A
(ii) (AB)¢ = B¢A¢
(iii) (kA)¢ = (kA¢).
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Q28Written answer
If then verify that:
(i) (2A + B)¢ = 2A¢ + B¢
(ii) (A – B)¢ = A¢ – B¢.
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Q29Written answer
Show that A¢A and AA¢ are both symmetric matrices for any matrix A.
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Worked solution
Let P = A¢A
So, P¢ = (A¢A) ¢
= A¢(A¢)¢ [As (AB) ¢ = B¢A¢]
Hence, A¢A is the symmetric matrix for any matrix A.
Now, let Q = AA¢
So, Q¢ = (AA¢)¢ = (A) ¢ = AA¢ = Q
Hence, AA¢ is the symmetric matrix for any matrix A.
Q30Written answer
Let A and B be square matrices of the order 3 × 3. Is (AB)2 = A2 B2? Give reasons.
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Worked solution
As, A and B be square matrices of order 3 x 3.
We have, (AB)2 = AB . AB
= A(BA)B
= A(AB)B [If AB = BA]
= AABB
= A2B2
Thus, (AB)2 = A2B2 is true only if AB = BA.