Matrix Addition Explained: Definition, Formula, Examples, and Practice Problems
Matrix addition is a basic operation in linear algebra. This guide provides definitions, formulas, examples, common mistakes, and practice exercises. All practice problems include hidden answers that you can click to reveal.
1. What Is a Matrix?
A matrix is a rectangular array of numbers arranged in rows and columns. Matrix size is written as rows × columns. Only matrices with the same dimensions can be added.
Example:
$$ A = \begin{bmatrix} 2 & 5 & -1 \\ 0 & 3 & 4 \end{bmatrix} $$This is a 2 × 3 matrix.
2. Definition and Formula of Matrix Addition
Matrix addition means adding two matrices element by element.
If
$$ A = [a_{ij}], \qquad B = [b_{ij}] $$Then
$$ A + B = [\,a_{ij} + b_{ij}\,] $$3. Example Problems
Example 1
Compute:
$$ A = \begin{bmatrix} 1 & 3 \\ 2 & 5 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & -1 \\ 0 & 7 \end{bmatrix} $$Solution:
$$ A + B = \begin{bmatrix} 1+4 & 3+(-1) \\ 2+0 & 5+7 \end{bmatrix} = \begin{bmatrix} 5 & 2 \\ 2 & 12 \end{bmatrix} $$Example 2
Compute:
$$ A = \begin{bmatrix} -3 & 6 & 2 \\ 1 & 0 & -4 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & -2 & 1 \\ -1 & 3 & 7 \end{bmatrix} $$Solution:
$$ A + B = \begin{bmatrix} -3+5 & 6+(-2) & 2+1 \\ 1+(-1) & 0+3 & -4+7 \end{bmatrix} = \begin{bmatrix} 2 & 4 & 3 \\ 0 & 3 & 3 \end{bmatrix} $$4. Common Mistakes to Avoid
- Adding matrices with different dimensions
- Incorrect handling of negative numbers
- Misaligning elements (adding the wrong positions)
- Confusing matrix addition with matrix multiplication
5. Practice Problems (Matrix Addition)
Click to reveal answers:
Practice 1
$$ \begin{bmatrix} 2 & 7 \\ 1 & -3 \end{bmatrix} + \begin{bmatrix} 5 & -2 \\ 0 & 4 \end{bmatrix} $$Practice 2
$$ \begin{bmatrix} -1 & 3 & 0 \\ 2 & 1 & 5 \end{bmatrix} + \begin{bmatrix} 4 & 6 & -2 \\ 3 & -1 & 0 \end{bmatrix} $$Practice 3
$$ \begin{bmatrix} 10 & -5 \\ 8 & 3 \end{bmatrix} + \begin{bmatrix} -3 & 1 \\ 6 & -2 \end{bmatrix} $$Practice 4
$$ \begin{bmatrix} 7 & 2 & -1 \\ 0 & 3 & 4 \end{bmatrix} + \begin{bmatrix} -4 & 5 & 1 \\ 6 & 1 & -3 \end{bmatrix} $$After matrix addition, you may also want to explore upper triangular form, RREF, and eigenvalues.
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