A matrix can be defined as an array of numbers, expressions that are arranged in rows and columns. A matrix is written in brackets [ ]. Dimension m*n refers to the total number of rows (m) and columns (n). An individual element in a matrix is termed as the entry.
The above matrix has 2 rows and 3 columns
Matrix multiplication can be categorized into two types.
- Scalar: It is one of the most elementary form of a matrix, wherein a single digit is multiplied by individual entry of a matrix.
- Multiplication: It is a multiplication of whole matrix by another whole matrix.
Check for Multiplication of Matrices
Before we go ahead with the steps for the multiplication of matrix with the other, it is required to check whether a multiplication between the two is possible. Let us take two matrices A and B for example. Matrix A can be multiplied with matrix B only if the number of columns in A is equal to the number of rows in B. So if m x n is the dimension of A, then the dimension of B must be n x p to be multiplied by A. And the product matrix, say C, will have the dimension m x p.
Steps for Multiplication of Matrices
The steps for multiplication of matrices are as follows:
- Check if the multiplication between the two matrices is possible by the rule stated above.
- If possible, multiply the first element of the first row of the first matrix to the first element of the first column of the second matrix, the second element of the first row of the first matrix to the second element of the first column of the second matrix and so on.
- Calculate and enter the result in as the first element of the first row of the result matrix.
- Next, multiply the first element of the first row of the first matrix to the first element of the second column of the second matrix, the second element of the first row of the first matrix to the second element of the second column of the second matrix and so on.
- Calculate and enter the result in as the second element of the first row of the result matrix.
- Now repeat the same steps with the second-row elements of the first matrix and keep entering the answers in the result matrix.
- Repeat for other rows and columns.
Multiplication by another Matrix Example
Consider Matrix A and Matrix B.
Now multiply matrices A and B
Multiply each element present along a first matrix row A with the corresponding elements present in the first row of B and sum the results.
Multiply the elements along the first rows of a column A with elements down the second column of a matrix B.
Multiplication proceeds along rows and columns as stated below
Scalar Matrix Multiplication Example
Matrix Multiplication Rules
- Ensure that number of columns present in the first matrix equals the number of rows present in the second matrix.
- Multiply the components present in the each row of a first matrix with the components of the each column present in the second matrix.
Matrix Vector Multiplication
In Order to multiply a row vector by a column vector, the number of columns present in the row vector must have same number of columns as that of columns present in the column vector. Following example illustrates the matrix vector multiplication.
One can find the applications of matrix in various fields. They are used in the field of computer graphics, physics and geometry.