Adding Matrix Solver
The Matrix Addition Solver is a tool designed to perform matrix addition effortlessly and accurately. It simplifies the process of adding two matrices by automatically checking their dimensions and calculating the sum of corresponding elements.
How to Use the Matrix Addition Solver
Input Matrices: Enter both matrices into the solver in a grid format.
Click Solve: Press the “Add” button to compute the sum.
View Results: The solver will display the resulting matrix along with the detailed computation.
Matrix Addition
Matrix addition is a straightforward operation where corresponding elements of two matrices are added together. For this to work, the matrices must have the same dimensions (i.e., the same number of rows and columns). If the dimensions do not match, the matrices cannot be added.
The rule is simple
\[
(A + B)_{ij} = A_{ij} + B_{ij}
\]
where \(A_{ij}\) and \(B_{ij}\) represent the elements in the \(i^{th}\) row and \(j^{th}\) column of matrices \(A\) and \(B\), respectively.
Example 1: Adding Two 2×2 Matrices
Add the following matrices:
\[
A = \begin{bmatrix} 2 & 3 \\ 5 & 7 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & 1 \\ 0 & 6 \end{bmatrix}
\]
Solution
Step 1: Add corresponding elements.
\[
\begin{aligned}
C_{11} &= A_{11} + B_{11} = 2 + 4 = 6 \\\\
C_{12} &= A_{12} + B_{12} = 3 + 1 = 4 \\\\
C_{21} &= A_{21} + B_{21} = 5 + 0 = 5 \\\\
C_{22} &= A_{22} + B_{22} = 7 + 6 = 13
\end{aligned}
\]
Step 2: Write the result as a new matrix:
\[
C = \begin{bmatrix} 6 & 4 \\ 5 & 13 \end{bmatrix}
\]
Example 2: Adding Two 3×3 Matrices
Add the following matrices:
\[
X = \begin{bmatrix} 1 & 0 & 3 \\ 4 & 2 & 5 \\ 6 & 7 & 8 \end{bmatrix}, \quad Y = \begin{bmatrix} 9 & 8 & 7 \\ 6 & 5 & 4 \\ 3 & 2 & 1 \end{bmatrix}
\]
Solution
Step 1: Add corresponding elements.
\[
\begin{aligned}
Z_{11} &= X_{11} + Y_{11} = 1 + 9 = 10, & Z_{12} &= X_{12} + Y_{12} = 0 + 8 = 8, & Z_{13} &= X_{13} + Y_{13} = 3 + 7 = 10 \\\\
Z_{21} &= X_{21} + Y_{21} = 4 + 6 = 10, & Z_{22} &= X_{22} + Y_{22} = 2 + 5 = 7, & Z_{23} &= X_{23} + Y_{23} = 5 + 4 = 9 \\\\
Z_{31} &= X_{31} + Y_{31} = 6 + 3 = 9, & Z_{32} &= X_{32} + Y_{32} = 7 + 2 = 9, & Z_{33} &= X_{33} + Y_{33} = 8 + 1 = 9
\end{aligned}
\]
Step 2: Write the result as a new matrix:
\[
Z = \begin{bmatrix} 10 & 8 & 10 \\ 10 & 7 & 9 \\ 9 & 9 & 9 \end{bmatrix}
\]
Key Reminder
Always double-check the dimensions before adding matrices. If they aren’t the same, addition is not defined!
