Last Updated on March 21, 2024 by Maged kamel

Matrix operations-part-1, addition, and subtraction.

What is the orthogonal matrix?

Orthogonal means perpendicular an example of two orthogonal matrices is given for the value of matrix A and its transpose. For a given matrix A, written as ( cos θ sinθ, -sinθ cosθ) Its transpose can be formed by changing the rows into columns, the transpose will be ( cos θ -sinθ, sinθ cosθ). When multiplying both matrices we will get an identity matrix.

When do two matrices said to be equal? if we have for instance matrix B. Then matrix A= matrix B, if each element of matrix A is equal to each element of matrix B. For a given matrix A, if the value of  a₁₁=a, then the corresponding b₁₁=3.
Both matrices A and B are (2×3), then matrix A= matrix B, if a11=b11=3, also a₁₂=-2=b₁₂ which is x. when a₁₁=b₁₁=3, and also when a₁₃=b=b13=-4, also if a₂₂=c = b₂₂ =-1, If c=-1.

For these conditions matrix, A= matrix B, or there will be equity between the two matrices.

matrix operations-part-1.

Matrix addition & subtraction.

What are matrix operations? adding matrices is the first operation of matrix operations part 1.

The first operation is adding two matrices or subtracting from each other. If we have a matrix A, which is (2×3) as shown, and another matrix B.

Both matrices must have (2×3), then we can perform the addition of these two matrices, A+B is the new matrix from addition, and each element is determined from the addition of the similar elements in matrix A and matrix B.

The first row with the first column=a₁₁₊b₁₁=1+1=2 will be the new(A+B) matrix first row of the first column. Similarly, a₂₁+b₂₁=2+1=3, which will be the new element for the second row with the first column.

The final matrix is shown due to the addition of A+B. For the value of the second row with the third column =(4+(-4)=0. The dimension f matrix (A+B) must be (2×3).

While subtracting matrix B from matrix A is the same as for the process of addition except that, we add A+( -1*B) as if we multiplied matrix B by (-1),thenู (-1) the new elements of (-B) will be as(-1*1,+2*-1,1*-1, 1*-1, 3*-1, -4*-1).

We add to the corresponding elements in matrix A. The final form of (A-B) is shown.

As (0,-4,2,1,-4, 8). Here is a list of the operations that can be done for matrices. The first item is adding A+B, which will be=B+A.

If we have three matrices A, B, C, we will add all these three matrices adding A+(B+C) will give the same result as adding (A+B)+(C). Adding A+(-A)=0, which is logic, since all elements will be zeros. Adding A+0 will give matrix=A.

Scalar multiplication.

 Scalar multiplication is the second operation of matrix operations part-1, scalar multiplication is multiplying matrix A by a constant, the new matrix will be multiple of the constant number. For the shown matrix 3×3, we want to multiply by 2, then the new matrix will have each element multiplied by 2, or it will be twice the old matrix before multiplying.

Matrix multiplication.

There is another operation that is Matrix multiplication, which is the third operation of matrix operations part-1

We draw an arrow from left to right and another arrow from top to bottom. We must consider that the number of columns in the first matrix is the same as the number of rows in the other matrix. Multiplying a matrix of dimension(1×3) with another matrix (3×1). The 1×3 means one row with three columns,3 is common, the first 3 are the column numbers and the other 3 are the row numbers.

The dimension f the final matrix will be( 1×1)which is the product of the first element of matrix A by the last column dimension in the other matrix.

The final matrix dimension is to be(1×1), or only one element. The final number is equal to (2*1+3*4+5*-6), and added together will give 16. The new matrix dimension will be(1×1).

For a useful external link, math is fun for the matrix part.
This is a link to the next post, Matrix operations part -2.