# Subtraction of Two Matrices – Definition, Conditions, Properties | How to Subtract Two Matrices?

Subtraction of Two Matrices is subtracting two matrices having the same order. The order must be the same for the two matrices. If the first matrix in the subtraction is 2 × 2, then the second matrix order must also be 2 × 2. The subtraction of matrices has different properties.

Learn the entire article and know how to subtract two matrices. To learn subtraction of matrices along with subtraction of two matrices, read 10th Grade Math articles available on our website. We have also provided 3 × 3 matrix subtraction, 4 × 4 matrix subtraction, etc.

## What is Subtraction of Two Matrices?

The subtraction of two matrices is the subtraction operation performed on two matrices. If A = [aij] and B = [bij] are two matrices with the same order or dimension and also have the same number of rows and a same number of columns, then the subtraction of matrices A and B becomes: A – B = [aij] – [bij] = [aij – bij]. To subtract two matrices, they must have the same order.

### How to Subtract Matrices?

You can subtract two matrices using the below procedure. Also, you can easily subtract any two matrices of any order with the given process.

• In the beginning, check whether the given matrices are having the same order or not.
• If the given matrices have the same dimension or order, then check out the elements having the same position.
• Subtract the elements of the first matrix with the respective elements of the second matrix.
• Then finally write the output of the subtraction of two matrices.

### Two Matrix Subtraction Examples | Problems on Subtracting Two Matrices

Check out the examples of Two Matrix Subtraction. All the given problems on Subtraction of Two Matrices will help the students to learn subtraction of matrices easily.

Example 1. If $$A =\left[ \begin{matrix} 5&4 \cr 6&3 \cr 9&7 \cr \end{matrix} \right]$$ and $$B =\left[ \begin{matrix} 10&8 \cr 17&1 \cr 2&4 \cr \end{matrix} \right]$$, then find the subtraction of A and B.

Solution:
Given matrices are $$A =\left[ \begin{matrix} 5&4 \cr 6&3 \cr 9&7 \cr \end{matrix} \right]$$ and $$B =\left[ \begin{matrix} 10&8 \cr 17&1 \cr 2&4 \cr \end{matrix} \right]$$
Both matrices have the same order of 3 × 2. Now, subtract the elements of the B matrix with the respective elements of the A matrix.
$$\left[ \begin{matrix} 5&4 \cr 6&3 \cr 9&7 \cr \end{matrix} \right]$$ – $$\left[ \begin{matrix} 10&8 \cr 17&1 \cr 2&4 \cr \end{matrix} \right]$$ = $$\left[ \begin{matrix} 5 – 10&4 – 8 \cr 6 – 17&3 – 1 \cr 9 – 2&7 – 4 \cr \end{matrix} \right]$$ = $$\left[ \begin{matrix} -5&-4 \cr -11&2 \cr 7&3 \cr \end{matrix} \right]$$

Therefore, the subtraction of A and B is $$\left[ \begin{matrix} -5&-4 \cr -11&2 \cr 7&3 \cr \end{matrix} \right]$$

Example 2. Subtract X and Y where $$X =\left[ \begin{matrix} 3&8 \cr 2&-5 \cr \end{matrix} \right]$$ and $$Y =\left[ \begin{matrix} -4&-3 \cr 2&1 \cr \end{matrix} \right]$$

Solution:
Given matrices are $$X =\left[ \begin{matrix} 3&8 \cr 2&-5 \cr \end{matrix} \right]$$ and $$Y =\left[ \begin{matrix} -4&-3 \cr 2&1 \cr \end{matrix} \right]$$
Both matrices have the same order of 2 × 2. Now, subtract the elements of the first matrix with the respective elements of the second matrix.
$$\left[ \begin{matrix} 3&8 \cr 2&-5 \cr \end{matrix} \right]$$ – $$\left[ \begin{matrix} -4&-3 \cr 2&1 \cr \end{matrix} \right]$$ = $$\left[ \begin{matrix} 3 + 4&8 + 3 \cr 2 – 2&-5 – 1 \cr \end{matrix} \right]$$ = $$\left[ \begin{matrix} 7&11 \cr 0&-6 \cr \end{matrix} \right]$$

Therefore, the subtraction of X and Y is $$\left[ \begin{matrix} 7&11 \cr 0&-6 \cr \end{matrix} \right]$$

Example 3.
Find a and b where the subtraction of $$A = \left[ \begin{matrix} 4 + a&2 \cr b + 6&2 \cr \end{matrix} \right]$$ and $$B = \left[ \begin{matrix} 8&2 \cr 12&2 \cr \end{matrix} \right]$$ is $$\left[ \begin{matrix} 1&0 \cr 3&0 \cr \end{matrix} \right]$$

Solution:
Given that $$A = \left[ \begin{matrix} 4 + a&2 \cr b + 6&2 \cr \end{matrix} \right]$$ and $$B = \left[ \begin{matrix} 8&2 \cr 12&2 \cr \end{matrix} \right]$$ is $$\left[ \begin{matrix} 1&0 \cr 3&0 \cr \end{matrix} \right]$$
Now, subtract the elements of A and B.
$$A – B = \left[ \begin{matrix} 4 + a – 8&2 – 2 \cr b + 6 – 12&2- 2 \cr \end{matrix} \right]$$ = $$\left[ \begin{matrix} 1&0 \cr 3&0 \cr \end{matrix} \right]$$
So, a – 4 = 1; a = 1 + 4; a = 5.
b – 6 = 3; b = 6 + 3; b = 9.

Therefore, the values of a and b are a = 5, and b = 9.

Example 4.
Subtract $$\left[ \begin{matrix} 3&4&9 \cr 12&14&13 \cr \end{matrix} \right]$$ and $$\left[ \begin{matrix} 5&2&7 \cr 10&11&9 \cr \end{matrix} \right]$$

Solution:
Given matrices are $$\left[ \begin{matrix} 3&4&9 \cr 12&14&13 \cr \end{matrix} \right]$$ and $$\left[ \begin{matrix} 5&2&7 \cr 10&11&9 \cr \end{matrix} \right]$$
Given matrices are 2 × 3 matrices of the same order. Now, Subtract the elements of the first matrix with the respective elements of the second matrix.
$$\left[ \begin{matrix} 3&4&9 \cr 12&14&13 \cr \end{matrix} \right]$$ – $$\left[ \begin{matrix} 5&2&7 \cr 10&11&9 \cr \end{matrix} \right]$$ =$$\left[ \begin{matrix} 3 – 5&4 – 2&9 – 7 \cr 12 – 10&14 – 11&13 – 9 \cr \end{matrix} \right]$$ = $$\left[ \begin{matrix} -2&2&2 \cr 2&3&4 \cr \end{matrix} \right]$$

Therefore, the required matrix is $$\left[ \begin{matrix} -2&2&2 \cr 2&3&4 \cr \end{matrix} \right]$$

### FAQs on Difference of Two Matrices

1. What is Subtraction of Two Matrices?

The subtraction taking place between two matrices is known as the Subtraction of Two Matrices.

2. How to Subtract two matrices A and B?

If you have two matrices such as A and B, then check their dimension or order first. If their order or dimension is the same, then subtract the elements of the same positions in the matrices.
If matrix A = aij and matrix B = bij, then the subtraction of A and B becomes A – B = aij – bij.

3. Can we subtract the 3×3 matrix from the 4×4 matrix?

No, it is not possible to subtract the 3×3 matrix from the 4×4 matrix. Because the main rule to subtract the matrices is that both matrices must be in the same order.

4. How to do subtraction for 2×2 matrices?

We can do the subtraction of the elements of 2 × 2 matrices of X and Y in the following order. That is
x11 – y11
x12 – y12
x22 – y22
x21 – y21

5. Does Commutative Law applicable for Subtraction of Matrices?

No, the Commutative Law is not applicable for Subtraction of Matrices. A – B ≠ B – A.

### Conclusion

At last, you will get to know what is the difference between the Subtraction of Two Matrices and also Subtraction of many Matrices. Begin your preparation from the Subtraction of Two Matrices. Then, it becomes easy to solve 3 × 3 matrices subtraction, 4 × 4 matrices subtraction, etc. Therefore, start preparing now and score well in the exam.

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