Subtraction of Matrices

Subtraction of matrices is addition of the negative of a matrix to another matrix which means A – B = A + (-B). The subtraction of the matrix is subtracting the corresponding row-column element of one matrix with same row-column element of another matrix.

In this article we will explore subtraction of matrices in detail. We will also solve some examples related to subtraction of matrices. Let’s start our learning on the topic “Subtraction of Matrices.”

Matrix subtraction is an operation where corresponding elements of two matrices are subtracted from each other to form a new matrix. This operation is similar to matrix addition, but instead of adding the corresponding elements, they are subtracted.

For two matrices to be subtracted, they must have the same dimensions, meaning they must have the same number of rows and columns. If A and B are two matrices of the same dimensions, their subtraction is denoted as A − B.

Notation of Matrix Subtraction

Given two matrices A and B of the same size, the matrix subtraction A−B is defined as:

(A − B)_ij ​= A_ij​ − B_ij​

where A_ij​ and B_ij​ are the elements of matrices A and B at the i^th row and j^th column, respectively.

Condition for Matrix Subtraction

For matrix subtraction to be defined and valid, the following condition must be satisfied:

The two matrices must have the same dimensions, meaning they must have the same number of rows and the same number of columns.

The subtraction of n × n matrices include subtracting each row column (i, j) element of one matrix with corresponding row column (i, j) element of other matrix.

Consider matrix P = [Tex] \begin{bmatrix} p_{11} & p_{12}& … &p_{1n}\\ p_{21} & p_{22}&… & p_{2n}\\ .&.&…& .\\ .&.&…& .\\ p_{n1}&p_{n2}&…&p_{nn} \end{bmatrix} [/Tex] and Q = [Tex] \begin{bmatrix} q_{11} & q_{12}& … &q_{1n}\\ q_{21} & q_{22}&… & q_{2n}\\ .&.&…& .\\ .&.&…& .\\ q_{n1}&q_{n2}&…&q_{nn} \end{bmatrix} [/Tex]

P – Q = [Tex] \begin{bmatrix} p_{11}-q_{11} & p_{12}-q_{12}& … &p_{1n}-q_{1n}\\ p_{21}-q_{21} & p_{22}-q_{22}&… & p_{2n}-q_{2n}\\ .&.&…& .\\ .&.&…& .\\ p_{n1}-q_{n1}&p_{n2}-q_{n2}&…&p_{nn}-q_{nn} \end{bmatrix} [/Tex]

Subtraction of 2 × 2 Matrices

The subtraction of 2 × 2 matrices includes subtracting each row column (i, j) element of one matrix with corresponding row column (i, j) element of other matrix.

Consider matrix X = [Tex]\begin{bmatrix} x_{11} & x_{12}\\ x_{21} & x_{22} \end{bmatrix}[/Tex] and Y = [Tex]\begin{bmatrix} y_{11} & y_{12}\\ y_{21} & y_{22} \end{bmatrix}[/Tex]

X – Y = [Tex]\begin{bmatrix} x_{11}- y_{11}& x_{12} – y_{12}\\ x_{21}-y_{21} & x_{22} – y_{22} \end{bmatrix}[/Tex]

Subtraction of 3 × 3 Matrices

The subtraction of 3 × 3 matrices includes subtracting each row column (i, j) element of one matrix with corresponding row column (i, j) element of other matrix.

Consider matrix X = [Tex]\begin{bmatrix} x_{11} & x_{12}&x_{13}\\ x_{21} & x_{22}&x_{23}\\ x_{31} & x_{32}&x_{33} \end{bmatrix}[/Tex] and Y = [Tex]\begin{bmatrix} y_{11} & y_{12}&y_{13}\\ y_{21} & y_{22}&y_{23}\\ y_{21} & y_{22}&y_{23}\\ \end{bmatrix}[/Tex]

X – Y = [Tex]\begin{bmatrix} x_{11}- y_{11}& x_{12} – y_{12}& x_{13} – y_{13}\\ x_{21}-y_{21} & x_{22} – y_{22}& x_{23} – y_{23}\\ x_{31}-y_{31} & x_{32} – y_{32}& x_{33} – y_{33} \end{bmatrix}[/Tex]

Read More,

• Matrices
• Matrix Operations
• Types of Matrices
• Order of Matrix
• Matrix Multiplication

Example 1: Q1. Subtract the matrices: P = [Tex]\begin{bmatrix} 15 & 4\\13 & 9 \end{bmatrix}[/Tex] and Q = [Tex]\begin{bmatrix} 1 & -13\\ 6 & 0 \end{bmatrix}[/Tex]

Solution:

P – Q = [Tex]\begin{bmatrix} 15 & 4\\13 & 9 \end{bmatrix}[/Tex] – [Tex]\begin{bmatrix} 1 & -13\\ 6 & 0 \end{bmatrix}[/Tex]

P – Q = [Tex]\begin{bmatrix} 15-1 & 4-(-13)\\ 13- 6 & 9-0 \end{bmatrix}[/Tex]

P – Q = [Tex]\begin{bmatrix} 14 & 17\\7 & 9 \end{bmatrix}[/Tex]

Example 2: Subtract the matrices A = [Tex]\begin{bmatrix} -8& 3& 9\\ -1 & 2&6\\ 5& -12&4 \end{bmatrix}[/Tex] and B = [Tex]\begin{bmatrix} -3& 5& 13\\ -10 & 23&0\\ 7& -1&22 \end{bmatrix}[/Tex]

Solution:

A – B = [Tex]\begin{bmatrix} -8& 3& 9\\ -1 & 2&6\\ 5& -12&4 \end{bmatrix}[/Tex] – [Tex]\begin{bmatrix} -3& 5& 13\\ -10 & 23&0\\ 7& -1&22 \end{bmatrix}[/Tex]

A – B = [Tex]\begin{bmatrix} -8-( -3)& 3-5& 9-13\\ -1-(-10) & 2-23&6-0\\ 5-7& -12-(-1)&4 -22 \end{bmatrix} [/Tex]

A – B = [Tex]\begin{bmatrix} -5& -2& -4\\ 9 & -21&6\\ -2& -11& -18 \end{bmatrix}[/Tex]

Example 3: Compute R – S where R = [Tex]\begin{bmatrix} -2& 5& 1&10\\ -3 & 7&-1&8\\ 6& -10&0&-4 \end{bmatrix}[/Tex] and S = [Tex]\begin{bmatrix} 8& 4& -3&0\\ -5 & -2&7&12\\ 0& 1&-2&9 \end{bmatrix}[/Tex]

Solution:

R – S = [Tex]\begin{bmatrix} -2& 5& 1&10\\ -3 & 7&-1&8\\ 6& -10&0&-4 \end{bmatrix}[/Tex] – [Tex]\begin{bmatrix} 8& 4& -3&0\\ -5 & -2&7&12\\ 0& 1&-2&9 \end{bmatrix}[/Tex]

R – S = [Tex]\begin{bmatrix} -2-8& 5-4& 1-(-3)&10-0\\ -3-(-5) & 7-(-2)&-1-7&8-12\\ 6-0& -10-1&0-(-2)&4-9 \end{bmatrix} [/Tex]

R – S = [Tex]\begin{bmatrix} -10& 1& 4&10\\ 2 & 9&-8&-4\\ 6& -11&2&-5 \end{bmatrix}[/Tex]

Example 4: Subtract the matrices C = [Tex]\begin{bmatrix} -1& 3 \\ 0 & 2\\ \end{bmatrix}[/Tex] and D = [Tex]\begin{bmatrix} 1& 3& 2\\ 4 & 5&9\\ 6& 10&11 \end{bmatrix}[/Tex]

Solution:

Order of C = 2 × 2 and Order of D = 3 × 3

Order of C ≠ Order of D

So, subtraction cannot be performed in the given matrices as the order of the given matrices are different and subtraction can only be performed on order of same matrices.

Q1. Subtract the matrices: P = [Tex]\begin{bmatrix} 5 & 9\\2 & 4 \end{bmatrix}[/Tex] and Q = [Tex]\begin{bmatrix} 0 & -3\\ 7 & 14 \end{bmatrix}[/Tex]

Q2. Subtract the matrices A = [Tex]\begin{bmatrix} -8& 3& 9\\ -1 & 2&6\\ 5& -12&4 \end{bmatrix}[/Tex] and B = [Tex]\begin{bmatrix} -3& 5& 13\\ -10 & 23&0\\ 7& -1&22 \end{bmatrix}[/Tex]

Q3. Compute R – S where R = [Tex]\begin{bmatrix} -8& 3& 9&10\\ -1 & 2&6&2\\ 5& -12&4&7 \end{bmatrix}[/Tex] and S = [Tex]\begin{bmatrix} 1& 6& 3&0\\ -1 & -2&-8&2\\ 0& 5&4&-7 \end{bmatrix}[/Tex]

Q4. Subtract the matrices C = [Tex]\begin{bmatrix} -18& 30 \\ -10 & 21\\ \end{bmatrix}[/Tex] and D = [Tex]\begin{bmatrix} -1& 3& 12\\ -2 & 4&6\\ 5& -3&14 \end{bmatrix}[/Tex]

What are the Rules for Subtracting Matrices?

The rules for subtracting matrices are:

• The matrices should be of equal dimensions.
• Each element of one matrix should be subtracted from corresponding element of other matrix.

Can You Subtract 2 × 2 Matrix from a 3 × 3 Matrix?

No, we cannot subtract 2 × 2 matrix from a 3 × 3 matrix as dimensions are not same.

How to Do Subtraction of Two Matrices?

To subtract two matrices, we subtract x_ij element of first matrix with y_ij of second matrix. This process is done for all the elements of the matrices.

Is Matrix Subtraction Associative?

No, matrix subtraction is not associative.

Is Matrix Subtraction Commutative?

No, matrix subtraction is not commutative.