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Transpose and Adjoint of Matrices

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 A^{T} . 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  A=\begin {bmatrix} a_{11} & a_{12} &a_{13} \\ a_{21} &a_{22} &a_{23} \\ a_{31} &a_{32} &a_{33} \end {bmatrix} If A = | a_{ij} | of order m*n then  A^{T} = | a_{ij} | of order n*m. So,  A^{T}=\begin {bmatrix} a_{11} & a_{21} &a_{31} \\ a_{12} &a_{22} &a_{32} \\ a_{13} &a_{23} &a_{33} \end {bmatrix} .

Example 1: Consider the matrix  A= \begin {bmatrix}2 & 3 &4 \\ 1 & -1 & 3 \end{bmatrix} . 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:  B=\begin {bmatrix} 2 & 1\\ 3 &-1 \\ 4 &3 \end {bmatrix} . The matrix B is called the transpose of A.

Example 2: Consider the matrix  A=\begin {bmatrix} 0 &7 &5 \\ 3 &8 &4 \\ 9 &6 &1 \end {bmatrix} . Do the transpose of matrix.

Solution: The transpose of matrix A by interchanging rows and columns is  A^{T}=\begin {bmatrix} 0 &3 &9 \\ 7 &8 &6 \\ 5 &4 &1 \end {bmatrix} .

Properties of Transpose

  1. The transpose of the transpose of a matrix is that the matrix itself =  (A^{T})^{T} = A
  2. The transpose of the addition of 2 matrices is similar to the sum of their transposes =  (A+B)^{T} = A^{T} + B^{T}
  3. When a scalar matrix is being multiplied by the matrix, the order of transpose is irrelevant =  (sA)^{T} = sA^{T}
  4. The transpose of the product of 2 matrices is similar to the product of their transposes in reversed order =  (AB)^{T} = B^{T}A^{T}

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,  A=\begin {bmatrix} A_{11} &A_{12} \\ A_{21}& A_{22} \end {bmatrix} then Adj A =  \begin {bmatrix} A_{11} &A_{21} \\ A_{12}& A_{22} \end {bmatrix}

Example 1: Consider the matrix  A=\begin {bmatrix} 5 &-1 \\ 2&2 \end {bmatrix} Find the Adj of A.

Solution: First to find out the minor and cofactor of the matrix :  M_{11} = 2  C_{11} = 2,  M_{12} = 2  C_{12} = -2,  M_{21} = -1  C_{21} = +1,  M_{22} = 5  C_{22} = 5.

Cofactor matrix =   \begin {bmatrix} 2 &-2 \\ 1&5 \end {bmatrix} and Adj A =    \begin {bmatrix} 2 &1 \\ -2&5 \end {bmatrix}

Example 2: Consider the matrix  C=\begin {bmatrix} 2 &-3 &-1 \\ 6&4&1 \\ 0&5&3 \end {bmatrix}  Find the Adj of A.

Solution:  M_{11} = 7 C_{11} = 7,  M_{12} = 18 C_{12} = -18,  M_{13} = 30 C_{13} = 30, M_{21} = 1 C_{21} = -1,  M_{22} = 6 C_{22} = 6,  M_{23} = 10 C_{23} = -10,  M_{31} = 1 C_{31} = 1,  M_{32} = 8 C_{32} = -8,  M_{33} = 26 C_{33} = 26.

Cofactor matrix =  \begin {bmatrix} 7 & -18 &30 \\ -1& 6 & -10\\ 1& -8 &26 \end{bmatrix} and Adj A =  \begin {bmatrix} 7 & -1 &1 \\ -18& 6 & -8\\ 30& -10 &26 \end{bmatrix} .

Exercise

  1. Find the adjoint of the matrix A=\begin {bmatrix} -1 &-2 &-2 \\ 2&1&-2 \\ 2&-2&1 \end {bmatrix} .
  2. Find the adjoint of matrix F=\begin {bmatrix} 4 &2 &3 \\ 4&0&1 \\ 1&1&0 \end {bmatrix} .
  3. Find the adjoint of matrix A=\begin {bmatrix} 8 &-9 \\ -5&6 \end {bmatrix} .
  4. Find the adjoint of matrix D=\begin {bmatrix} 4 &-5 \\ 2&1 \end {bmatrix} .
  5. Find the adjoint of the matrix G=\begin {bmatrix} 3 &-4 &1 \\ -3&6&-1 \\ 4&-6&2 \end {bmatrix} .


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