To determine the values of [tex]\( m \)[/tex] and [tex]\( n \)[/tex] in the given matrix addition problem, we need to add the corresponding elements of the two matrices on the left side of the equation and set them equal to the corresponding elements in the resulting matrix on the right side.
Given the equation:
[tex]\[
\left[\begin{array}{cc}
n-1 & 6 \\
-19 & m+3
\end{array}\right]+\left[\begin{array}{cc}
-1 & 0 \\
16 & -8
\end{array}\right]=\left[\begin{array}{cc}
10 & 6 \\
-3 & 40
\end{array}\right]
\][/tex]
Let's add the matrices element by element:
1. For the element in the first row, first column:
[tex]\[
(n - 1) + (-1) = 10
\][/tex]
Simplifying:
[tex]\[
n - 1 - 1 = 10
\][/tex]
[tex]\[
n - 2 = 10
\][/tex]
Adding 2 to both sides:
[tex]\[
n = 12
\][/tex]
2. For the element in the first row, second column:
[tex]\[
6 + 0 = 6
\][/tex]
This agrees with the element in the resulting matrix.
3. For the element in the second row, first column:
[tex]\[
-19 + 16 = -3
\][/tex]
This agrees with the element in the resulting matrix.
4. For the element in the second row, second column:
[tex]\[
(m + 3) + (-8) = 40
\][/tex]
Simplifying:
[tex]\[
m + 3 - 8 = 40
\][/tex]
[tex]\[
m - 5 = 40
\][/tex]
Adding 5 to both sides:
[tex]\[
m = 45
\][/tex]
Therefore, the values of [tex]\( m \)[/tex] and [tex]\( n \)[/tex] are:
[tex]\(
m = 45
\)[/tex]
[tex]\(
n = 12
\)[/tex]
