Examples Of Properties Of Determinants

Determinants can be demonstrated as scalar values used to perform various mathematical calculations. The matrix and determinants are similar in many ways; only the difference is that the set of values in the matrix is written under the brackets, while the set of values in the determinant is written under the bars. The number of rows and columns in the matrix is not equal every time. But the determinant number of rows and columns remains equal in almost every case. The matrix multiplication is slightly different from the multiplication of determinants. 

Matrix and determinants

Let’s learn about the matrix and determinants.

Matrix and determinants both are interrelated to each other. They both are used to perform various mathematical calculations. The matrix represents numbers in the rows and columns, while the matrix is the values obtained after solving rows and columns. 

Value of determinant 

The value of the determinant is obtained from the matrix. For finding the value of the matrix, one row or column is taken. After that, the first element of that particular row or column is taken. Then, the elements in the row or column of that element get neglected, and all the elements are computed by performing cross multiplication. Similar action is taken by all the elements of that row or column. After that, the computed values are added or subtracted accordingly. The addition and subtraction are performed based on the Power of 1, obtained by adding the value of row and column. After performing all the computations, the resulting value is the determinant of the matrix. 

Properties of matrix and determinants with examples 

Let’s understand some properties of matrix and determinants. 

• Whether the matrix is calculated using the row or column, the value of the determinant will remain the same. Although, it will remain the same after taking any matrix element. For example, if there is a matrix with 3 elements in columns 2,5 and 3. Then whether the value of the determinant is computed using the 2 or 5, it will remain the same. 
• If the value of all elements in the rows and columns is zero, then the matrix multiplication will also be zero. Although, the value of its determinant will also be zero. For example, if matrix A has all the values 0,0,0 and…. Then the value of matrix multiplication is zero. Along with this, the determinant value id is also zero. 
• Suppose the matrix is an identity matrix. The identity matrix is one whose all the diagonals have a value of 1, and other elements are zero. Then the matrix multiplication of such a matrix will give zero value. However, the value of their determinant will also be zero. For example, suppose any 3 × 3 is an identity matrix, and all elements are zero except diagonals. Its matrix multiplication will be zero, and the value of its determinant is also zero. 
• If the elements of rows and columns in any matrix get changed, then the value of the determinant will not change, but its sign will change. For example, if the matrix has 2 elements in columns 5 and 5 and two in n rows, 6 and 6. Then if 6 and 5 shifted to columns, or 5 and 5 shifted to row, the sign of the determinant’s value changed. 
• Suppose the elements in the rows and columns of any matrix are the same. Then the value of their determinant will be zero. For example, if the columns have elements 5 and 5, rows also have two elements 5 and 5. Then the value of their determinant will be zero. 

Matrix squared

Let’s understand the matrix squared in detail. 

The matrix squared is also called the square matrix. The square matrix is that matrix whose number of rows and columns remains the same. The order of the matrix squared will remain the same, and any operation like matrix multiplication, division, subtraction and addition can be performed easily.

Conclusion

Matrix and determinants are used in performing various derivations of mathematics and physics. It is also used in other fields like cryptography, photography, etc. The value of the determinant depends on the matrix. If the value of matrix elements gets changed, then the matrix’s value also gets changed. Although, according to the properties of matrix and determinants, if the elements of rows and columns get changed, the sign of their determinant also changes. Although, if the determinant is computed using any value of the given matrix, then its value remains constant.