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

Transcript text: If $a$ and $b$ are nonzero real numbers and $A=\left[\begin{array}{cc}a b & b^{2} \\ -a^{2} & -a b\end{array}\right]$. Find $A^{2}$

Solution

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.

Step 1: Define the Matrix $ A $

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

Step 2: Calculate $ A^2 $

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

Step 3: Perform Matrix Multiplication

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

The matrix $ A^2 $ is: \[ \boxed{\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}} \]

Was this solution helpful?

Unhelpful

Helpful

Questions asked by other users

< Previous Next >

Thank you for your feedback.

Copied!