Determine if B is the inverse matrix of A by calculating A B...

Determine if $B$ is the inverse matrix of $A$ by calculating $A B$ and $B A$
$$
A=\left[\begin{array}{ll}
4 & 3 \\
5 & 4
\end{array}\right], \quad B=\left[\begin{array}{rr}
4 & -3 \\
-5 & 4
\end{array}\right]
$$

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Key Concepts

Matrix Inverse

A matrix inverse is defined for a square matrix A as a matrix B such that the product of A with B (in both orders) results in the identity matrix. This concept is crucial in solving systems of linear equations and understanding how to 'undo' the effect of a matrix transformation.

Matrix Multiplication

Matrix multiplication is the process of combining two matrices to produce a third matrix, where each element in the result is obtained as the sum of products of corresponding elements from rows of the first matrix and columns of the second. It is key in verifying whether one matrix can act as the inverse of another through multiplication.

Identity Matrix

The identity matrix is a special kind of square matrix characterized by ones on its main diagonal and zeros elsewhere. It serves as the multiplicative identity in matrix operations, meaning that any matrix multiplied by the identity matrix remains unchanged. In the context of matrix inverses, achieving the identity matrix upon multiplication confirms the inverse relationship.

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