What is the value of $A$ in the matrix equation below?
$A+\left[\begin{array}{cccc}
3 & 9 & -1 & -8 \\
16 & -2 & 3 & 13
\end{array}\right]=\left[\begin{array}{cccc}
0 & -5 & 6 & 10 \\
3 & 0 & -2 & 7
\end{array}\right]$
A. $\left[\begin{array}{cccc}-3 & -14 & 7 & 18 \\
19 & 2 & -5 & -6\end{array}\right]$
B. $\left[\begin{array}{cccc}-3 & -14 & 7 & 18 \\
-13 & 2 & -5 & -6\end{array}\right]$
C. $\left[\begin{array}{cccc}-3 & -14 & 7 & 2 \\
-13 & 2 & -5 & -6\end{array}\right]$
D. $\left[\begin{array}{cccc}-3 & -14 & 7 & 2 \\
19 & 2 & -5 & -6\end{array}\right]$
Answer (1)
Subtract matrix B from matrix C to find matrix A: A = C − B .
Calculate each element of A by subtracting corresponding elements: A ij = C ij − B ij .
Compute the elements: A 11 = − 3 , A 12 = − 14 , A 13 = 7 , A 14 = 18 , A 21 = − 13 , A 22 = 2 , A 23 = − 5 , A 24 = − 6 .
The resulting matrix A is: [ − 3 − 14 7 18 − 13 2 − 5 − 6 ] .
Explanation
Problem Analysis We are given the matrix equation A + B = C , where we need to find the matrix A . Here, B = [ 3 9 − 1 − 8 16 − 2 3 13 ] and C = [ 0 − 5 6 10 3 0 − 2 7 ] .
Finding Matrix A To find the matrix A , we need to subtract matrix B from matrix C . This can be written as A = C − B . We perform this subtraction element-wise.
Calculating Elements of A The elements of matrix A are calculated as follows:
A 11 = C 11 − B 11 = 0 − 3 = − 3
A 12 = C 12 − B 12 = − 5 − 9 = − 14
A 13 = C 13 − B 13 = 6 − ( − 1 ) = 6 + 1 = 7
A 14 = C 14 − B 14 = 10 − ( − 8 ) = 10 + 8 = 18
A 21 = C 21 − B 21 = 3 − 16 = − 13
A 22 = C 22 − B 22 = 0 − ( − 2 ) = 0 + 2 = 2
A 23 = C 23 − B 23 = − 2 − 3 = − 5
A 24 = C 24 − B 24 = 7 − 13 = − 6
Final Answer Therefore, the matrix A is:
A = [ − 3 − 14 7 18 − 13 2 − 5 − 6 ]
Examples
Matrix subtraction is used in image processing to find the difference between two images. For example, if you have two images of the same scene taken at different times, subtracting the matrices representing the images can highlight changes that occurred in the scene between the two times. This technique is used in surveillance and security systems to detect movement or changes in a monitored area. The matrix elements represent pixel intensities, and the subtraction reveals the differences.
