If 𝐴 = [−2 43 −1] 𝑎𝑛𝑑 𝐵 = [1 34 −1], then verify that either B is a multiplicative inverse ofA or not?

1. Break Down the Problem

To determine if matrix $B$ is the multiplicative inverse of matrix $A$, we need to check if the product of $A$ and $B$ results in the identity matrix.

2. Relevant Concepts

For $A$ and $B$ to be inverses, the following must hold true: $A \times B = I$ where $I$ is the identity matrix. The identity matrix $I$ for a $3 \times 3$ matrix is: $I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

3. Analysis and Detail

Let's define the matrices:

$A = \begin{bmatrix} -2 & 43 & -1 \\ \end{bmatrix} , \quad B = \begin{bmatrix} 1 & 34 & -1 \\ \end{bmatrix}$

Now, we compute the product $A \times B$:

$A \times B = \begin{bmatrix} -2 & 43 & -1 \end{bmatrix} \times \begin{bmatrix} 1 \\ 34 \\ -1 \\ \end{bmatrix}$

Calculating the dot product:

$= (-2 \times 1) + (43 \times 34) + (-1 \times -1)$ $= -2 + 1462 + 1$ $= 1461$

4. Verify and Summarize

The product $A \times B$ gives us a single scalar rather than the identity matrix, which indicates that $B$ is not the inverse of $A$.

Final Answer

Thus, $B$ is not a multiplicative inverse of $A$.