
Adjoint of the matrix N=$\left[ \begin{matrix} -4 & -3 & -3 \\ 1 & 0 & 1 \\ 4 & 4 & 3 \\ \end{matrix} \right]$ is [MP PET 1989]
E. N
F. 2N
G. - N
H. None of these
Answer
161.1k+ views
Hint: To answer the adjoint of a matrix question, we must first identify the cofactor of each matrix element. Create a new matrix using the cofactors, then expand the cofactors to produce the matrix. Then, transpose the matrix you determined in the previous step.
Formula Used: The following equation can be used to determine the cofactor for a specific element: $Aij = (-1)^{i+j} det M_{ij}$
Complete step by step solution:Given N=$\left[ \begin{matrix} -4 & -3 & -3 \\ 1 & 0 & 1 \\ 4 & 4 & 3 \\ \end{matrix} \right]$
We will first evaluate each element's cofactor,
So, the cofactors of N are
${{c}_{11}}=-4,\,{{c}_{12}}=1,\,{{c}_{13}}=4$
${{c}_{21}}=-3,\,{{c}_{22}}=0,\,{{c}_{23}}=4$
${{c}_{31}}=-3,\,{{c}_{32}}=1,\,{{c}_{33}}=3$
Therefore, the transpose of the cofactor matrix is an Adjoint matrix.
N = $\left[ \begin{matrix} {{c}_{11}} & {{c}_{12}} & {{c}_{13}} \\ {{c}_{21}} & {{c}_{22}} & {{c}_{23}} \\ {{c}_{31}} & {{c}_{32}} & {{c}_{33}} \\ \end{matrix} \right]$=$\left[ \begin{matrix} 1 & 2 & -2 \\ 2 & 5 & -4 \\ 3 & 7 & -5 \\ \end{matrix} \right]$
Hence, the adjoint matrix formed is:
$adj\,N=\left[ \begin{matrix} -4 & -3 & -3 \\ 1 & 0 & 1 \\ 4 & 4 & 3 \\ \end{matrix} \right]$=N
Option ‘A’ is correct
Note: In a matrix—a cofactor is a number that is obtained by removing the row and column of a particular element. Generally, the cofactor is preceded by a positive (+) or negative (-) sign. Once the co-factor members of a matrix are transposed, the adjoint of the matrix is formed.
Formula Used: The following equation can be used to determine the cofactor for a specific element: $Aij = (-1)^{i+j} det M_{ij}$
Complete step by step solution:Given N=$\left[ \begin{matrix} -4 & -3 & -3 \\ 1 & 0 & 1 \\ 4 & 4 & 3 \\ \end{matrix} \right]$
We will first evaluate each element's cofactor,
So, the cofactors of N are
${{c}_{11}}=-4,\,{{c}_{12}}=1,\,{{c}_{13}}=4$
${{c}_{21}}=-3,\,{{c}_{22}}=0,\,{{c}_{23}}=4$
${{c}_{31}}=-3,\,{{c}_{32}}=1,\,{{c}_{33}}=3$
Therefore, the transpose of the cofactor matrix is an Adjoint matrix.
N = $\left[ \begin{matrix} {{c}_{11}} & {{c}_{12}} & {{c}_{13}} \\ {{c}_{21}} & {{c}_{22}} & {{c}_{23}} \\ {{c}_{31}} & {{c}_{32}} & {{c}_{33}} \\ \end{matrix} \right]$=$\left[ \begin{matrix} 1 & 2 & -2 \\ 2 & 5 & -4 \\ 3 & 7 & -5 \\ \end{matrix} \right]$
Hence, the adjoint matrix formed is:
$adj\,N=\left[ \begin{matrix} -4 & -3 & -3 \\ 1 & 0 & 1 \\ 4 & 4 & 3 \\ \end{matrix} \right]$=N
Option ‘A’ is correct
Note: In a matrix—a cofactor is a number that is obtained by removing the row and column of a particular element. Generally, the cofactor is preceded by a positive (+) or negative (-) sign. Once the co-factor members of a matrix are transposed, the adjoint of the matrix is formed.
Recently Updated Pages
If there are 25 railway stations on a railway line class 11 maths JEE_Main

Minimum area of the circle which touches the parabolas class 11 maths JEE_Main

Which of the following is the empty set A x x is a class 11 maths JEE_Main

The number of ways of selecting two squares on chessboard class 11 maths JEE_Main

Find the points common to the hyperbola 25x2 9y2 2-class-11-maths-JEE_Main

A box contains 6 balls which may be all of different class 11 maths JEE_Main

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Displacement-Time Graph and Velocity-Time Graph for JEE

Degree of Dissociation and Its Formula With Solved Example for JEE

Free Radical Substitution Mechanism of Alkanes for JEE Main 2025

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced 2025: Dates, Registration, Syllabus, Eligibility Criteria and More

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations

NCERT Solutions for Class 11 Maths In Hindi Chapter 1 Sets

NCERT Solutions for Class 11 Maths Chapter 6 Permutations and Combinations
