1. What is a matrix?

A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are commonly used in linear algebra, computer graphics, data science, and many applications of engineering and physics. A matrix with m rows and n columns is called an m × n matrix.

Example of a general \(m\times n\) matrix: \[ A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} \]

2. Matrix subtraction: definition and formula

Matrix subtraction is defined only for two matrices of the same dimensions. If \(A = [a_{ij}]\) and \(B = [b_{ij}]\) are both \(m\times n\) matrices, then their difference is

\[ A - B = [\,a_{ij} - b_{ij}\,] \quad\text{for } 1 \le i \le m,\; 1 \le j \le n. \]

Important: If the matrices have different numbers of rows or columns, subtraction is not defined.

3. Worked examples

4. Common mistakes to avoid

• Not checking dimensions: Always verify both matrices have the same number of rows and columns before subtracting.
• Reversing the order: Matrix subtraction is not commutative: \(A - B \ne B - A\) in general.
• Mis-handling negative signs: Remember \(a - (-b) = a + b\).
• Reading the wrong entry: For larger matrices double-check row and column indices (e.g., \(a_{23}\) is row 2, column 3).

5. Practice problems (try them first — answers are hidden)

1. Problem 1
   \[ \begin{bmatrix} 4 & 7 \\ 1 & 3 \end{bmatrix} - \begin{bmatrix} 2 & 1 \\ 5 & 0 \end{bmatrix} \]
   Show answer

   \[ \begin{bmatrix} 4-2 & 7-1 \\ 1-5 & 3-0 \end{bmatrix} = \begin{bmatrix} 2 & 6 \\ -4 & 3 \end{bmatrix} \]
2. Problem 2
   \[ \begin{bmatrix} -1 & 6 & 2 \\ 0 & 3 & -4 \end{bmatrix} - \begin{bmatrix} 2 & -2 & 5 \\ 1 & 0 & -1 \end{bmatrix} \]
   Show answer

   Subtract element-wise: \[ \begin{bmatrix} -1-2 & 6-(-2) & 2-5 \\ 0-1 & 3-0 & -4-(-1) \end{bmatrix} = \begin{bmatrix} -3 & 8 & -3 \\ -1 & 3 & -3 \end{bmatrix} \]
3. Problem 3
   \[ \begin{bmatrix} 10 & 0 \\ -3 & 7 \\ 5 & 1 \end{bmatrix} - \begin{bmatrix} 4 & 2 \\ 1 & -5 \\ 0 & 3 \end{bmatrix} \]
   Show answer

   \[ \begin{bmatrix} 10-4 & 0-2 \\ -3-1 & 7-(-5) \\ 5-0 & 1-3 \end{bmatrix} = \begin{bmatrix} 6 & -2 \\ -4 & 12 \\ 5 & -2 \end{bmatrix} \]
4. Problem 4 (slightly harder)
   \[ \begin{bmatrix} 3 & -1 & 4 \\ 2 & 8 & 0 \\ -5 & 1 & 7 \end{bmatrix} - \begin{bmatrix} 1 & 3 & -2 \\ -2 & 0 & 6 \\ 4 & -3 & 1 \end{bmatrix} \]
   Show answer

   Subtract each corresponding entry: \[ \begin{bmatrix} 3-1 & -1-3 & 4-(-2) \\ 2-(-2) & 8-0 & 0-6 \\ -5-4 & 1-(-3) & 7-1 \end{bmatrix} = \begin{bmatrix} 2 & -4 & 6 \\ 4 & 8 & -6 \\ -9 & 4 & 6 \end{bmatrix} \]

6. Quick tips for using a matrix subtraction calculator

• Always verify the input matrix sizes shown on the page before computing.
• If your calculator accepts CSV or pasted rows, make sure you separate rows consistently (commas, spaces or new lines as required).
• For teaching, present subtraction step-by-step: show original matrices side-by-side and highlight corresponding entries.