## 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

## 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 =

**Example 1: **Consider the matrix Find the Adj of A.

**Solution: **First to find out the minor and cofactor of the matrix : = 2 = 2, = 2 = -2, = -1 = +1, = 5 = 5.

Cofactor matrix = and Adj A =

**Example 2: **Consider the matrix Find the Adj of A.

**Solution:** = 7 = 7, = 18 = -18, = 30 = 30, = 1 = -1, = 6 = 6, = 10 = -10, = 1 = 1, = 8 = -8, = 26 = 26.

Cofactor matrix = and Adj A = .

## Exercise

- Find the adjoint of the matrix .
- Find the adjoint of matrix .
- Find the adjoint of matrix.
- Find the adjoint of matrix .
- Find the adjoint of the matrix .

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