Question

If A = \left( {\begin{array}{*{20}{c}} 2&{ - 3} \\\ 4&1 \end{array}} \right), then the adjoint of the matrix $A$ is __________.
A) \left( {\begin{array}{*{20}{c}} 1&3 \\\ { - 4}&2 \end{array}} \right)
B) \left( {\begin{array}{*{20}{c}} 1&{ - 3} \\\ { - 4}&2 \end{array}} \right)
C) \left( {\begin{array}{*{20}{c}} 1&{ - 3} \\\ 4&{ - 2} \end{array}} \right)
D) \left( {\begin{array}{*{20}{c}} { - 1}&{ - 3} \\\ { - 4}&2 \end{array}} \right)

Answer 1

To find the adjoint of the given matrix we need to find the cofactor of the elements of the given matrix.
In the $2 \times 2$ matrix the opposite element is the cofactor of the respective element.
Adjoint of a $2 \times 2$ matrix is nothing but the transpose of cofactor of the respective matrix.

Complete step-by-step answer:
To find the adjoint of the given matrix we are first in need to find the cofactor of the elements of the given matrix.
From the given hint
The cofactor of $2$ is said to be $1$.
The cofactor of $- 3$ is said to be $- 4$.
The cofactor of $4$ is said to be $3$.
The cofactor of $1$ is said to be $2$.
Now we should take the transpose of these cofactors in order to arrive at the adjoint of the given matrix.
That means the columns will be rows and the rows will be columns.
So,
Cofactor of A = \left( {\begin{array}{*{20}{c}} 1&{ - 4} \\\ 3&2 \end{array}} \right), it is denoted as cof(A) = \left( {\begin{array}{*{20}{c}} 1&{ - 4} \\\ 3&2 \end{array}} \right)
Transpose of cofactor of A is {(cof(A))^T} = \left( {\begin{array}{*{20}{c}} 1&3 \\\ { - 4}&2 \end{array}} \right)
From the definition of adjoint of a $2 \times 2$matrix we know that
Adj A = \left( {\begin{array}{*{20}{c}} 1&3 \\\ { - 4}&2 \end{array}} \right)
Hence, from the above found adjoint of A we have come to an conclusion to that the correct option is (A) \left( {\begin{array}{*{20}{c}} 1&3 \\\ { - 4}&2 \end{array}} \right).

Note: The adjoint of a matrix (also called the adjugate of a matrix) is defined as the transpose of the cofactor matrix of that particular matrix. For a matrix A, the adjoint is denoted as adj (A).
The transpose of a matrix is a new matrix whose rows are the columns of the original. (This makes the columns of the new matrix the rows of the original).