Questions: If a and b are nonzero real numbers and A=[ab b^2 / -a^2 -ab]. Find A^2

Solution Steps

To find \( A^2 \), we need to multiply matrix \( A \) by itself. This involves calculating the dot product of rows and columns of the matrix.

Given the matrix: \[ A = \begin{bmatrix} ab & b^2 \\ -a^2 & -ab \end{bmatrix} \]

To find \( A^2 \), compute the matrix multiplication \( A \times A \): \[ A^2 = \begin{bmatrix} ab & b^2 \\ -a^2 & -ab \end{bmatrix} \times \begin{bmatrix} ab & b^2 \\ -a^2 & -ab \end{bmatrix} \]

Calculate each element of \( A^2 \):

• First row, first column: \( (ab)(ab) + (b^2)(-a^2) = a^2b^2 - a^2b^2 = 0 \)
• First row, second column: \( (ab)(b^2) + (b^2)(-ab) = ab^3 - ab^3 = 0 \)
• Second row, first column: \( (-a^2)(ab) + (-ab)(-a^2) = -a^3b + a^3b = 0 \)
• Second row, second column: \( (-a^2)(b^2) + (-ab)(-ab) = -a^2b^2 + a^2b^2 = 0 \)

Thus, the resulting matrix is: \[ A^2 = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \]

Final Answer