Math Doubts

Subtraction of Matrices

A mathematical approach of subtracting a matrix from another matrix is called subtraction of matrices.

Matrices are sometimes involved in subtraction. The subtraction of them can be done by subtracting elements in one matrix from the corresponding elements of another matrix. Remember, it is possible to perform the subtraction of matrices only when they are of the same order.

Example

$A = $ $\begin{bmatrix} 7 & 1 \\
0 & 2 \\
\end{bmatrix}$ and $B = $ $\begin{bmatrix} 5 & 6 \\
3 & 9 \\
\end{bmatrix}$ are two matrices.

The matrices $A$ and $B$ are second order matrices. Therefore, every matrix has four elements. Now, subtract matrix $B$ from matrix $A$ and it is written as $A-B$ in mathematics.

$\implies A-B$ $\,=\,$ $\begin{bmatrix} 7 & 1 \\
0 & 2 \\
\end{bmatrix}$ $-$ $\begin{bmatrix} 5 & 6 \\
3 & 9 \\
\end{bmatrix}$

Subtract each element of matrix $B$ from the corresponding element of matrix $A$.

$\implies A-B$ $\,=\,$ $\begin{bmatrix} 7-5 & 1-6 \\
0-3 & 2-9 \\
\end{bmatrix}$

$\,\,\, \therefore \,\,\,\,\,\, A-B$ $\,=\,$ $\begin{bmatrix} 2 & -5 \\
-3 & -7 \\
\end{bmatrix}$

The subtraction of matrices is successfully completed because of the subtraction of same order matrices. As demonstrated, the subtraction of two matrices is performed in mathematics.

Now, try to subtract matrices which belong to different orders and it understands us the difficulty of subtracting matrices whose orders are different.

$C = $ $\begin{bmatrix} 2 & 4 & 7 \\
-5 & 6 & 0 \\
\end{bmatrix}$ and $D = $ $\begin{bmatrix} 1 & -4 \\
3 & 7 \\
2 & 5 \\
\end{bmatrix}$ are two matrices of different order.

The matrix $C$ is an order of $2 \times 3$ and $D$ is a matrix of order $3 \times 2$. The matrix $C$ contains six elements whereas the matrix $D$ also has six elements.

  1. $2$ and $1$ are elements belong to first row, first column of matrices $C$ and $D$ respectively. Subtract the first row, first column elements as a first step of subtracting the matrices.
  2. $4$ and $-4$ are elements belong to first row, second column of matrices $C$ and $D$ respectively. Subtract the first row, second column elements as a second step of subtracting the matrices.
  3. $7$ is an element of first row, third column of matrix $C$ but there is no element of matrix $D$ at that position. So, the matrix $C$ and $D$ cannot be subtracted further. So, the subtraction of matrices $C$ and $D$ is failed.

The subtraction of matrices of different order is failed in three steps in this example. In this way, the matrices are failed to subtract each other if their orders are different. Therefore, it proves that the matrices get subtracted only when they are of the same order.



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