Simplify (B⁻¹A⁻¹)⁻¹ given A = 2 & 2 −3 & 2 and B = 0 & −1 1 & 0 and choose the correct option.

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To find (B⁻¹A⁻¹)⁻¹, we use the property of matrix inverses (XY)⁻¹ = Y⁻¹X⁻¹. Step 1: Apply the inverse property to the given expression. Using the property (XY)⁻¹ = Y⁻¹X⁻¹, we can rewrite the expression: (B⁻¹A⁻¹)⁻¹ = (A⁻¹)⁻¹(B⁻¹)⁻¹ Step 2: Apply the property of double inverse. We know that (X⁻¹)⁻¹ = X. Applying this to our expression: (A⁻¹)⁻¹ = A; (B⁻¹)⁻¹ = B So, the expression simplifies to: (B⁻¹A⁻¹)⁻¹ = AB Step 3: Calculate the product of matrices A and B. Given matrices: A = 2 & 2 −3 & 2 B = 0 & −1 1 & 0 Now, we multiply A by B: AB = (2)(0) + (2)(1) & (2)(−1) + (2)(0) (−3)(0) + (2)(1) & (−3)(−1) + (2)(0) AB = 0 + 2 & −2 + 0 0 + 2 & 3 + 0 AB = 2 & −2 2 & 3 Step 4: Compare the result with the given options. The calculated matrix is 2 & −2 2 & 3 . This matches option (A). The final answer is A.

Simplify (B⁻¹A⁻¹)⁻¹ given A = 2 & 2 -3 & 2 and B = 0 & -1 1 & 0 and choose the correct option.

Answer

Step-by-step solution

Apply the inverse property to the given expression.

Apply the property of double inverse.

Calculate the product of matrices $A$ and $B$ .

Compare the result with the given options.

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