Matrices & Determinants
Grade 12
Question:
$A = \begin{bmatrix} 2 & -1 \\ -7 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 4 & 1 \\ 7 & 2 \end{bmatrix}$ then $B^T A^T$ is
a null matrix
an identity matrix
scalar, but not an identity matrix
such that $Tr(B^T A^T) = 4$
Step-by-Step Solution
Key Concept: The product $B^T A^T$ is equivalent to $(AB)^T$ according to the reversal law of transposes, and results in the identity matrix when $B$ is the inverse of $A$.
$A = \begin{bmatrix} 2 & -1 \\ -7 & 4 \end{bmatrix}, B = \begin{bmatrix} 4 & 1 \\ 7 & 2 \end{bmatrix} \Rightarrow A^T = \begin{bmatrix} 2 & -7 \\ -1 & 4 \end{bmatrix}, B^T = \begin{bmatrix} 4 & 7 \\ 1 & 2 \end{bmatrix}$. $B^T A^T$ is an identity matrix.
Correct Answer: B