Matrices and Linear Equations

If the number of rows and columns of a matrix are equal, then it is called a square matrix.

A matrix which is not square is called a rectangular matrix. Thus the number of rows and columns are not equal in a rectangular matrix.

The matrix of type n×n whose diagonal entries are 1 and the remaining entries are 0 is called the identity matrix of order n.

Let A= [a_ij] and B= [b_ij] be matrices of order m×n. If a_ij=b_ij for all 1 ≤i≤m and 1≤j≤n (that is, the corresponding entries are equal), these two matrices are said to be equal.

We just add all the components in the same row and column (i.e. a₁₁+b₁₁). So the sum will be:

No. The number of columns and rows of two matrices should be same to do the addition operation. 

In order to be able to multiply two matrices, the number of columns of the first matrix must be equal to the number of rows of the second matrix. For instance we can multiply a 4x3 matrix with 3x2 matrix, and output will be a 4x2 matrix. However we can not multiply a 4x3 matrix with a 2x3 matrix.

If a matrix A has an inverse, A is called invertible (or nonsingular); if such an inverse does not exist, A is called singular.

We have to multiply all the components of the matrix with 3.

Thus:

Thus AB is not equal to BA

We obtain three possible cases for the solution of this system: 1. There is no solution if the lines are parallel. There is only one solution if the lines intersect. 3.There are infinitely many solutions if they coincide.

detA=(1*0)-(2*-1)=0-(-2)=0+2=2

The determinant of singular matrices is equal to zero. For the given matrix,

det(A)=1*6-2*3=6-6=0. So matrix A is singular.

We dont have to multiply the matrices since det(AB)=det(A)*det(B).

det(A)=1*6-2*4=-2

det(B)=3*4-(-2)(-3)=12-6=6

So det(AB)=-2*6=-12

In order to write the equation system we have to first find the coefficient matrix. Then we write as CX=D where C is the coefficent matrix, X is the variable vector (x, y) and D is the right side of the equations.

Thus:

For an equation in the form ax+by=c, the slope of the line corresponding to the equation is -a/b and the x and y intercepts are c/a and c/b respectively. If the slopes of 2 lines are same and if they have different intercepts, then they are parallel, which in turn means the equation system has no solution. On the other hand, if the slopes and the intercepts are same then the lines coincide, which in turn means there are infinitely many solutions. Finally if the slopes are different then there is exactly one solution.

For the given equation system, slopes are same but intercepts are different. Thus they are parallel lines, which in turn means that the given equaion system has no solution.

We will use Sarrus' rule:

det(A+B)=1 since it is the identity matrix.

However det(A)+det(B)=(-2-3)+(-2-3)=-10