Last Updated on September 2, 2026 by Maged kamel
Matrix Operations-part-2.
In this new post, I will introduce matrix operations—part 2, which covers matrix powers. Please refer to the next slide image.
The data are quoted from Prof. Kuldeep Singh’s handbook, which discusses the properties of addition through a solved problem.
Powers of a matrix.
The following property of matrix operations, part 2, is the powers of the matrix. For instance, matrix A with dimension (2×2) consists of 4 elements (1 2, 0 1).
A raised to the power of 2, or A^2, can be done by multiplying matrix A by itself.
If matrix A has a dimension of (2×2), the final matrix will also have a dimension of (2×2).
Let us perform the multiplication. It is easy to consider the first row as a vertical column and multiply it by the first column of matrix A. For (Ab)11, we have (1*1 + 2*0) = 1 + 0 = 1. I use the letter B to indicate the second matrix.
For (Ab)12, we multiply the first row of A by the second column of B, and after placing it as a vertical column, we have (1*2+2*-1)=2+(-2)=0.
For (Ab)21, we multiply the second row, and after placing it as a vertical column by the first column of Matrix A, we have (0*2+-1*0)=0+0=0.
For (Ab)22, we multiply the second row, and after placing it as a vertical column by the second column of Matrix A, we have (0*2+-1*-1)=0+1=1. We cannot square each element of matrix A and consider the resulting matrix to be A^2.
Multiplication of 3×3 matrices.
For the matrix operations-part-2 of multiplying two matrices AxB, matrix A has a dimension of 3×3, while the second matrix B has a dimension of 3×4. Since both matrices have a standard dimension of 3 for the columns of A and the row of b, the final dimension of the product of A by b will be of the dimension of (3×4).
Elements AB11, AB12.
Consider the first row as a vertical column and multiply it by the first column of matrix B. For (AB)11, we have (3*2+5*5+-1*9)=22.
For (AB)12, we multiply the first row of A by the second column of B; we have (3*-2+5*0+1*-4)=-2.
Elements AB13, AB14.
For (AB)13, we multiply the first row, and after placing it as a vertical column by the third column of Matrix B, we have (3*3+5*7+(-1*1)=43.
For (AB)14, we multiply the first row, and after placing it as a vertical column by the fourth column of Matrix B, we have (3*1+5*8+(-1*1)=42.
Now, we can write down the first row for the product of matrix A by matrix B.
Elements AB21, AB22 & AB23.
For (AB)21, we multiply the second row and, after placing it as a vertical column to the left of the first column of Matrix B, we have (4*2+0*5+2*9)=8+18=26.
For (AB)22, we multiply the second row, and after placing it as a vertical column by the second column of Matrix B, we have (4*-2+0*0+2*-4)=-8-8=-16.
For (AB)23, we multiply the second row by the third column of Matrix B, and after placing the result in a vertical column, we have (4*3+0*7+2*1)=12+2=14.
Element AB24.
For (AB)24, we multiply the second row by the fourth column of Matrix B, and we get (4*1 + 0*8 + 2*1) = 6.
Now, we can write down the first & second rows of the product of matrices A and B.
Elements AB31, AB32& AB33 and AB34.
For (AB)31, we multiply the third row and, after placing it as a vertical column to the left of the first column of Matrix B, we have (-6*2+-3*5+2*9)=-12-15+18=-9.
For (AB)32, we multiply the third row, and after placing it as a vertical column by the second column of Matrix B, we have (-6*-2+-3*0+2*-4)=+12-0-8=4.
For (AB)33, we multiply the third row, and after placing it as a vertical column by the third column of Matrix B, we have (-6*3+-3*7+2*1)=-18-21+2=-37.
For (AB)34, we multiply the third row, and after placing it as a vertical column by the fourth column of Matrix B, we have (-6*1+-3*8+2*1)=-6-24+2=-28.
The Product of A by B is shown in the next slide image. We cannot compute the product of B by A because the number of columns in Matrix B is 4, while the number of rows in Matrix A is only 3. An online matrix calculator verifies the product of matrix A by matrix B.
The PDF data for this post and the next one can be viewed or downloaded via the next document.
This is the next post, 5-Solved problem for matrix operations.
For a useful external link, Math is Fun for the matrix part.