If A is a square matrix such that A^2 = A , then det(A) equals a.- 1 or 1 b.- 2 or 2 c.- 3 or 3 d.None of these

Answers 2

Option d) None of these ✔️

Answer:

[tex]\qquad\qquad\boxed{ \bf{ \: \: \:(d) \: \: \: None\:of\:these \: \: \: }} \\ \\ [/tex]

Explanation:

Given that,

[tex]\qquad\sf \: {A}^{2} = A \\ \\ [/tex]

Taking determinant on both sides, we get

[tex]\qquad\sf \: | {A}^{2} | = |A| \\ \\ [/tex]

can be rewritten as

[tex]\qquad\sf \: |A.A | = |A| \\ \\ [/tex]

[tex]\qquad\sf \: |A| |A| = |A| \\ \\ [/tex]

[tex]\qquad\sf \: |A| |A| - |A| = 0 \\ \\ [/tex]

[tex]\qquad\sf \: |A| (|A| -1)= 0 \\ \\ [/tex]

[tex]\qquad\sf \: |A| = 0 \: \: \: or \: \: \: |A| -1= 0 \\ \\ [/tex]

[tex]\qquad\sf\implies \sf \: |A| = 0 \: \: \: or \: \: \: |A| = 1\\ \\ [/tex]

[tex]\rule{190pt}{2pt}[/tex]

Additional Information

If A and B are square matrices of order n, then

[tex]\sf \: |AB| = |A| \: |B| \\ \\ [/tex]

[tex]\sf \: |adjA| = { |A| }^{n - 1} \\ \\ [/tex]

[tex]\sf \: |A \: adjA| = { |A| }^{n} \\ \\ [/tex]

[tex]\sf \: | {A}^{ - 1} | = { |A| }^{ - 1} \\ \\ [/tex]

[tex]\sf \: |A| = |A'| \\ \\ [/tex]

[tex]\sf \: |adj(adj \: A)| = { |A| }^{ {(n - 1)}^{2} } \\ \\ [/tex]