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Properties of Determinants

If B=B2 and...

Question

If $B = B^{2}$ and $I - B = A$ then ( $A, B$ are square matrices of same order)

A

A^{2} = B

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B

A^{2} = A

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C

A^{2} = I

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D

A^{2} = - A

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Solution

The correct option is B $A^{2} = A$
$B = B^{2}$ and $I - B = A$
$A^{2} = {(I - B)}^{2} = (I - B) (I - B) = I^{2} - B I - B I + B^{2} = I - 2 B + B^{2} = I - 2 B + B (∵ B^{2} = B) = I - B = A (∵ I - B = A) ∴ A^{2} = A$

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Q. If A and B are square matrices of the same order, then (A + B)(A − B) is equal to

(a) A² − B²
(b) A² − BA − AB − B²
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Q. If A and B are square matrices of the same order, explain, why in general
(i) (A + B)² ≠ A² + 2AB + B²
(ii) (A − B)² ≠ A² − 2AB + B²
(iii) (A + B) (A − B) ≠ A² − B².

Q. If AB = A and BA = B, where A and B are square matrices, then
(a) B² = B and A² = A
(b) B² ≠ B and A² = A
(c) A² ≠ A, B² = B
(d) A² ≠ A, B² ≠ B

A

B

are square matrices of the same order such that

(A + B) (A - B) = A^{2} - B^{2}

(A B A^{- 1})^{2}

A

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are square matrices of the same order such that

A^{2} = A, B^{2} = B, A B = B A = 0

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