Transpose and Adjoint of Matrices

Maths Tutor

7 years ago

Transpose of a Matrix

The matrix obtained from a given matrix A by interchanging its rows and columns is called Transpose of matrix A. Transpose of A is denoted by A’ or . If A is of order m*n, then A’ is of the order n*m. Clearly, the transpose of the transpose of A is the matrix A itself i.e. (A’)’= A.

Consider the matrix If A = || of order m*n then = || of order n*m. So, .

Example 1: Consider the matrix . Do the transpose of matrix.

Solution: It is an order of 2*3. By, writing another matrix B from A by writing rows of A as columns of B. We have: . The matrix B is called the transpose of A.

Example 2: Consider the matrix . Do the transpose of matrix.

Solution: The transpose of matrix A by interchanging rows and columns is .

Properties of Transpose

The transpose of the transpose of a matrix is that the matrix itself = = A
The transpose of the addition of 2 matrices is similar to the sum of their transposes =
When a scalar matrix is being multiplied by the matrix, the order of transpose is irrelevant =
The transpose of the product of 2 matrices is similar to the product of their transposes in reversed order =

Adjoint of a Matrix

Given a square matrix A, the transpose of the matrix of the cofactor of A is called adjoint of A and is denoted by adj A. An adjoint matrix is also called an adjugate matrix. In other words, we can say that matrix A is another matrix formed by replacing each element of the current matrix by its corresponding cofactor and then taking the transpose of the new matrix formed.

Suppose, then Adj A =