Answer
Verified
393.3k+ views
Hint: 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).
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).
Recently Updated Pages
Basicity of sulphurous acid and sulphuric acid are
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What is the stopping potential when the metal with class 12 physics JEE_Main
The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main
Using the following information to help you answer class 12 chemistry CBSE
Trending doubts
State the differences between manure and fertilize class 8 biology CBSE
Why are xylem and phloem called complex tissues aBoth class 11 biology CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
What would happen if plasma membrane ruptures or breaks class 11 biology CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
What precautions do you take while observing the nucleus class 11 biology CBSE
What would happen to the life of a cell if there was class 11 biology CBSE
Change the following sentences into negative and interrogative class 10 english CBSE