To determine which value of [tex]\( t \)[/tex] makes the two matrices inverses of each other, we need to check if their sum equals the identity matrix. The given matrices are:
[tex]\[
\begin{pmatrix}
-4 & 6 \\
3 & -4
\end{pmatrix}
\][/tex]
and
[tex]\[
\begin{pmatrix}
2 & 3 \\
1.5 & t
\end{pmatrix}.
\][/tex]
First, let's perform the addition of the two matrices:
[tex]\[
\begin{pmatrix}
-4 & 6 \\
3 & -4
\end{pmatrix}
+
\begin{pmatrix}
2 & 3 \\
1.5 & t
\end{pmatrix}
=
\begin{pmatrix}
-4 + 2 & 6 + 3 \\
3 + 1.5 & -4 + t
\end{pmatrix}
=
\begin{pmatrix}
-2 & 9 \\
4.5 & t - 4
\end{pmatrix}.
\][/tex]
For the matrices to be inverses of each other, their sum must be the identity matrix:
[tex]\[
\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}.
\][/tex]
Therefore, we need:
[tex]\[
\begin{pmatrix}
-2 & 9 \\
4.5 & t - 4
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}.
\][/tex]
Let's equate the corresponding elements of the matrices:
1. [tex]\(-2 = 1\)[/tex]
2. [tex]\(9 = 0\)[/tex]
3. [tex]\(4.5 = 0\)[/tex]
4. [tex]\(t - 4 = 1\)[/tex]
From the first three equations, [tex]\(-2 = 1\)[/tex], [tex]\(9 = 0\)[/tex], and [tex]\(4.5 = 0\)[/tex], it's clear that no values of [tex]\( t \)[/tex] can satisfy these equations. Matrices inverses are only making sense if the entries match, which they do not.
Therefore, no value of [tex]\( t \)[/tex] makes the sum of the two given matrices equal to the identity matrix. Hence, none of the given values of [tex]\( t \)[/tex] ([tex]\(-3, -2, 2, 3\)[/tex]) will result in the matrices being inverses of each other.
