Algebra

Matrices & Determinants

Medium

applied

If A is a square matrix such that $A^2 = A$ and I is the identity matrix of the same order as A, then $(I+A)^2-3A$ is equal to

✅ Correct Option: 1

Given that $A$ is a square matrix such that $A^2 = A$ and $I$ is the identity matrix of the same order as $A$ .

Expanding $(I+A)^2$ :

$(I+A)^2 = (I+A)(I+A)$

$= I \cdot I + I \cdot A + A \cdot I + A \cdot A$

$= I^2 + IA + AI + A^2$

Using the properties $I^2 = I$ , $IA = A$ , and $AI = A$ :

$(I+A)^2 = I + A + A + A^2$

$= I + 2A + A^2$

Substituting into the expression $(I+A)^2 - 3A$ :

$(I+A)^2 - 3A = I + 2A + A^2 - 3A$

$= I + A^2 - A$

Using the given condition $A^2 = A$ :

$I + A^2 - A = I + A - A$

$= I$

Therefore, $(I+A)^2 - 3A = I$

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(A) Scalar matrix

(B) Symmetric matrix

(D) Row matrix

Choose the correct answer from the options given below:

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Match List-I with List-II

List-I	List-II
(A). $AB - BA$	(I). Neither symmetric nor Skew symmetric
(B). $BA - 2AB$	(II). Skew symmetric
(C). $AB + BA$	(III). Null matrix
(D). $(AB)^T - BA$	(IV). Symmetric

Choose the correct answer from the options given below:

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If A is a square matrix such that A2=AA^2 = AA2=A and I is the identity matrix of the same order as A, then (I+A)2−3A(I+A)^2-3A(I+A)2−3A is equal to