GATE Overflow Test Series | Linear Algebra | Test 1 | Question: 8

Question

If $A$ is skew- symmetric matrix of order $3,$ then the matrix $A^{6}$ is?

• A. Diagonal matrix
• B. Skew-symmetric matrix
• C. Symmetric matrix
• D. None of the above

Answer 1

We have $A^{T} = -A$

$(A^{6})^{T} = (A\cdot A \cdot A \cdot A \cdot A \cdot A)^{T} = A^{T} \;A^{T} \;A^{T}\;A^{T}\;A^{T} \;A^{T} = (-A)(-A)(-A)(-A)(-A)(-A) = (-1)^{6}A^{6} = A^{6}.$

$\therefore$ The matrix $A^{6}$ is symmetric matrix.

So, the correct answer is $(C).$