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