To find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] for which the sum of matrices
[tex]\[
A = \begin{pmatrix}
2x - 1 & 5 \\
2y + 1 & 2
\end{pmatrix}
\][/tex]
and
[tex]\[
B = \begin{pmatrix}
2 & 2x - 5 \\
3 & x - 1
\end{pmatrix}
\][/tex]
will equal the identity matrix
[tex]\[
I = \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix},
\][/tex]
we need to set up and solve the equation [tex]\( A + B = I \)[/tex].
First, calculate the sum of matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
A + B = \begin{pmatrix}
2x - 1 & 5 \\
2y + 1 & 2
\end{pmatrix}
+ \begin{pmatrix}
2 & 2x - 5 \\
3 & x - 1
\end{pmatrix}
= \begin{pmatrix}
(2x - 1) + 2 & 5 + (2x - 5) \\
(2y + 1) + 3 & 2 + (x - 1)
\end{pmatrix}.
\][/tex]
Simplify each element in the resulting matrix:
[tex]\[
A + B = \begin{pmatrix}
2x - 1 + 2 & 5 + 2x - 5 \\
2y + 1 + 3 & 2 + x - 1
\end{pmatrix}
= \begin{pmatrix}
2x + 1 & 2x \\
2y + 4 & x + 1
\end{pmatrix}.
\][/tex]
We require this sum to equal the identity matrix [tex]\( I \)[/tex], therefore:
[tex]\[
\begin{pmatrix}
2x + 1 & 2x \\
2y + 4 & x + 1
\end{pmatrix}
= \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}.
\][/tex]
Now, equate the corresponding elements of the matrices:
1. [tex]\( 2x + 1 = 1 \)[/tex]
2. [tex]\( 2x = 0 \)[/tex]
3. [tex]\( 2y + 4 = 0 \)[/tex]
4. [tex]\( x + 1 = 1 \)[/tex]
Solve these equations step by step:
1. From [tex]\( 2x + 1 = 1 \)[/tex]:
[tex]\[
2x + 1 = 1 \implies 2x = 0 \implies x = 0.
\][/tex]
2. From [tex]\( 2x = 0 \)[/tex]:
[tex]\[
2x = 0 \implies x = 0.
\][/tex]
3. From [tex]\( 2y + 4 = 0 \)[/tex]:
[tex]\[
2y + 4 = 0 \implies 2y = -4 \implies y = -2.
\][/tex]
4. From [tex]\( x + 1 = 1 \)[/tex]:
[tex]\[
x + 1 = 1 \implies x = 0.
\][/tex]
Therefore, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that will make the sum of matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex] equal to the identity matrix are [tex]\( x = 0 \)[/tex] and [tex]\( y = -2 \)[/tex].
