Answer :
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.
[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.
