Consider the following for the two (02) items that follow: Let
\(\begin{vmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{vmatrix}\)
Understanding the Matrix Property A(adj A)
The question asks us to find the value of the expression \(A(\text{adj } A)\) for a given \(3 \times 3\) matrix \(A\).
We are given the matrix:
| \[A = \begin{vmatrix}3 & -3 & 4 \\2 & -3 & 4 \\0 & -1 & 1\end{vmatrix}\] |
A fundamental property relating a matrix \(A\), its adjugate matrix (\(\text{adj } A\)), and its determinant (\(|A|\)) is:
\[A(\text{adj } A) = (\text{adj } A)A = |A|I\]
where \(I\) is the identity matrix of the same order as \(A\).
To find \(A(\text{adj } A)\), we first need to calculate the determinant of matrix \(A\), denoted as \(|A|\).
Calculating the Determinant of Matrix A
The determinant of a \(3 \times 3\) matrix \(\begin{vmatrix}a & b & c \\d & e & f \\g & h & i\end{vmatrix}\) is calculated as \(a(ei - fh) - b(di - fg) + c(dh - eg)\).
For our matrix \(A = \begin{vmatrix}3 & -3 & 4 \\2 & -3 & 4 \\0 & -1 & 1\end{vmatrix}\), the determinant is:
\[|A| = 3 \begin{vmatrix}-3 & 4 \\-1 & 1\end{vmatrix} - (-3) \begin{vmatrix}2 & 4 \\0 & 1\end{vmatrix} + 4 \begin{vmatrix}2 & -3 \\0 & -1\end{vmatrix}\]
Let's compute the determinants of the \(2 \times 2\) matrices:
- \(\begin{vmatrix}-3 & 4 \\-1 & 1\end{vmatrix} = (-3)(1) - (4)(-1) = -3 - (-4) = -3 + 4 = 1\)
- \(\begin{vmatrix}2 & 4 \\0 & 1\end{vmatrix} = (2)(1) - (4)(0) = 2 - 0 = 2\)
- \(\begin{vmatrix}2 & -3 \\0 & -1\end{vmatrix} = (2)(-1) - (-3)(0) = -2 - 0 = -2\)
Now substitute these values back into the determinant formula for \(A\):
\[|A| = 3(1) - (-3)(2) + 4(-2)\]
\[|A| = 3 + 6 - 8\]
\[|A| = 9 - 8\]
\[|A| = 1\]
Applying the Matrix Property
We found that the determinant of matrix \(A\) is \(|A| = 1\). Using the property \(A(\text{adj } A) = |A|I\), we can substitute the value of \(|A|\):
\[A(\text{adj } A) = (1)I\]
Since \(A\) is a \(3 \times 3\) matrix, the identity matrix \(I\) must also be a \(3 \times 3\) identity matrix:
\[I = \begin{vmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{vmatrix}\]
Therefore, multiplying the identity matrix by 1 gives:
\[A(\text{adj } A) = 1 \cdot \begin{vmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{vmatrix} = \begin{vmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{vmatrix}\]
Final Result
The expression \(A(\text{adj } A)\) is equal to the \(3 \times 3\) identity matrix.
| \[\begin{vmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{vmatrix}\] |
Similar Questions
- Let X be a matrix of order \(3 \times 3\), Y be a matrix of order \(2 \times 3\) and Z be a matrix of order \(3 \times 2\). Which of the following statements are correct?
I. (ZY)X is defined and is a square matrix of order 3.
II. Y(XZ) is defined and is a square matrix of order 2.
III. X(YZ) is not defined.
Select the answer using the code given below. Let \(A\) and \(B\) be matrices of order \(3 \times 3\)., If \(|A| = \frac{1}{2\sqrt{2}}\) and \(|B| = \frac{1}{729}\), then what is the value of \(|2B(\text{adj}(3A))|\)?
- Consider the following statements in respect of two non-singular matrices \(A\) and \(B\) of the same order \(n\):
1. \(\text{adj}(AB) = (\text{adj}A)(\text{adj}B)\)
2. \(\text{adj}(AB) = \text{adj}(BA)\)
3. \((AB)\text{adj}(AB) - |AB|I_n\) is a null matrix of order \(n\)
How many of the above statements are correct? - Consider the following statements in respect of a non-singular matrix \(A\) of order \(n\):
1. \(A(\text{adj}A^T) = A(\text{adj}A)^T\)
2. If \(A^2 = A\), then \(A\) is identity matrix of order \(n\)
3. If \(A^3 = A\), then \(A\) is identity matrix of order \(n\)
Which of the statements given above are correct? - Consider the following statements in respect of a skew-symmetric matrix \(A\) of order \(3\):
1. All diagonal elements are zero.
2. The sum of all the diagonal elements of the matrix is zero.
3. \(A\) is orthogonal matrix.
Which of the statements given above are correct? If $A = \begin{bmatrix} \sin 2\theta & 2 \sin^2 \theta - 1 & 0 \\ \cos 2\theta & 2 \sin \theta \cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$, then which of the following statements is/are correct?
1. $A^{-1} = \text{adj}A$
2. $A$ is skew-symmetric matrix
3. $A^{-1} = A^T$
Select the correct answer using the code given below :- Let A and B be two square matrices of same order. If AB is a null matrix, then which one of the following is correct?
If
\(\begin{vmatrix}1 + \omega & 1 + \omega^2 & \omega + \omega^2 \\1 & \omega & \omega^2 \\\frac{1}{\omega} & \frac{1}{\omega^2} & 1\end{vmatrix}\)
\(\omega = -\frac{1}{2} + i\frac{\sqrt{3}}{2}\)
then what isequal to?
- If P is a skew-symmetric matrix of order 3, then what is det(P) equal to?
- If \(\begin{vmatrix} a-b & p-q & x-y \\ b-c & q-r & y-z \\ c-a & r-p & z-x \end{vmatrix} = k \begin{vmatrix} a & b & c \\ p & q & r \\ x & y & z \end{vmatrix}\) then what is the value of k ?
Important Questions from Matrices and Determinants
- If each element of a matrix is zero, then the matrix is:
- Let X be a matrix of order \(3 \times 3\), Y be a matrix of order \(2 \times 3\) and Z be a matrix of order \(3 \times 2\). Which of the following statements are correct?
I. (ZY)X is defined and is a square matrix of order 3.
II. Y(XZ) is defined and is a square matrix of order 2.
III. X(YZ) is not defined.
Select the answer using the code given below. Let \(A\) and \(B\) be matrices of order \(3 \times 3\)., If \(|A| = \frac{1}{2\sqrt{2}}\) and \(|B| = \frac{1}{729}\), then what is the value of \(|2B(\text{adj}(3A))|\)?
- Consider the following statements in respect of two non-singular matrices \(A\) and \(B\) of the same order \(n\):
1. \(\text{adj}(AB) = (\text{adj}A)(\text{adj}B)\)
2. \(\text{adj}(AB) = \text{adj}(BA)\)
3. \((AB)\text{adj}(AB) - |AB|I_n\) is a null matrix of order \(n\)
How many of the above statements are correct? - Consider the following statements in respect of a non-singular matrix \(A\) of order \(n\):
1. \(A(\text{adj}A^T) = A(\text{adj}A)^T\)
2. If \(A^2 = A\), then \(A\) is identity matrix of order \(n\)
3. If \(A^3 = A\), then \(A\) is identity matrix of order \(n\)
Which of the statements given above are correct?
