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What are [tex]\( C_{12} \)[/tex], [tex]\( C_{31} \)[/tex], and [tex]\( C_{22} \)[/tex] of matrix [tex]\( C \)[/tex]?

[tex]\[
C = \left[\begin{array}{cc}
16 & 9 \\
-3 & 0 \\
4 & -10
\end{array}\right] + \left[\begin{array}{cc}
-0.5 & 0 \\
5 & 8 \\
-3 & 14
\end{array}\right]
\][/tex]

Calculate the elements of the resulting matrix [tex]\( C \)[/tex] and identify the values of [tex]\( C_{12} \)[/tex], [tex]\( C_{31} \)[/tex], and [tex]\( C_{22} \)[/tex].



Answer :

GinnyAnswer
To find the elements [tex]\( C_{12}, C_{31}, \)[/tex] and [tex]\( C_{22} \)[/tex] of the resulting matrix [tex]\( C \)[/tex], we should first add the corresponding elements of the given matrices.

Given matrices:

[tex]\[ C_1 = \begin{bmatrix} 16 & 9 \\ -3 & 0 \\ 4 & -10 \end{bmatrix} \][/tex]
[tex]\[ C_2 = \begin{bmatrix} -0.5 & 0 \\ 5 & 8 \\ -3 & 14 \end{bmatrix} \][/tex]

The addition of these matrices is done element-wise to get matrix [tex]\( C \)[/tex]:

[tex]\[ C = C_1 + C_2 = \begin{bmatrix} 16 & 9 \\ -3 & 0 \\ 4 & -10 \end{bmatrix} + \begin{bmatrix} -0.5 & 0 \\ 5 & 8 \\ -3 & 14 \end{bmatrix} \][/tex]

[tex]\[ C = \begin{bmatrix} 16 + (-0.5) & 9 + 0 \\ -3 + 5 & 0 + 8 \\ 4 + (-3) & -10 + 14 \end{bmatrix} \][/tex]

Let's compute each element:

[tex]\[ \begin{bmatrix} 16 + (-0.5) & 9 + 0 \\ -3 + 5 & 0 + 8 \\ 4 + (-3) & -10 + 14 \end{bmatrix} = \begin{bmatrix} 15.5 & 9 \\ 2 & 8 \\ 1 & 4 \end{bmatrix} \][/tex]

Now, we can extract the required elements:

[tex]\[ C_{12} = \text{Element in the first row, second column} = 9 \][/tex]

[tex]\[ C_{31} = \text{Element in the third row, first column} = 1 \][/tex]

[tex]\[ C_{22} = \text{Element in the second row, second column} = 8 \][/tex]

Thus, the values are:

[tex]\[ C_{12} = 9.0 \][/tex]

[tex]\[ C_{31} = 1.0 \][/tex]

[tex]\[ C_{22} = 8.0 \][/tex]

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