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?
Adjugate Matrix Properties Analysis
This solution examines the properties of the adjugate matrix concerning the product of two non-singular matrices, \(A\) and \(B\), of the same order \(n\). We will evaluate three given statements based on fundamental principles of matrix algebra.
Matrix Properties Overview
- Non-Singular Matrix: A square matrix is non-singular if its determinant is non-zero. For such matrices, the inverse exists.
- Adjugate Matrix (\(\text{adj}(M)\)): The adjugate of a square matrix \(M\) is the transpose of its cofactor matrix.
- Fundamental Property: For any \(n \times n\) matrix \(M\), the product of the matrix and its adjugate is given by: \(M \cdot \text{adj}(M) = \text{adj}(M) \cdot M = |M| I_n\) where \(|M|\) is the determinant of \(M\) and \(I_n\) is the identity matrix of order \(n\).
- Determinant of a Product: For square matrices \(A\) and \(B\) of the same order, \(|AB| = |A||B|\).
- Adjugate of a Product: For non-singular matrices \(A\) and \(B\), the adjugate of their product is given by: \(\text{adj}(AB) = (\text{adj}B)(\text{adj}A)\) Note the reversal of the order compared to matrix multiplication.
Statement 1: Adjugate of Product Commutation
This statement proposes that the adjugate of the product \(AB\) is equal to the product of the adjugates \((\text{adj}A)(\text{adj}B)\).
However, the established property for the adjugate of a product of two matrices \(A\) and \(B\) is:
\(\text{adj}(AB) = (\text{adj}B)(\text{adj}A)\)The statement \(\text{adj}(AB) = (\text{adj}A)(\text{adj}B)\) is only true if \((\text{adj}B)(\text{adj}A) = (\text{adj}A)(\text{adj}B)\), which means the adjugates of \(A\) and \(B\) must commute. This is not generally true for arbitrary non-singular matrices \(A\) and \(B\). Therefore, Statement 1 is incorrect.
Statement 2: Adjugate Product Equality
This statement claims that the adjugate of the product \(AB\) is equal to the adjugate of the product \(BA\). This implies:
\((\text{adj}B)(\text{adj}A) = (\text{adj}A)(\text{adj}B)\)As established previously, this equality requires that the adjugates of \(A\) and \(B\) commute. Since \(A\) and \(B\) are arbitrary non-singular matrices, their adjugates \((\text{adj}A)\) and \((\text{adj}B)\) do not necessarily commute. For example:
Let \(A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\) and \(B = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}\).
We calculate:
- \(AB = \begin{pmatrix} 3 & 2 \\ 7 & 4 \end{pmatrix}\)
- \(BA = \begin{pmatrix} 1 & 2 \\ 4 & 6 \end{pmatrix}\)
- \(\text{adj}(A) = \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix}\)
- \(\text{adj}(B) = \begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix}\)
- \(\text{adj}(AB) = \begin{pmatrix} 4 & -2 \\ -7 & 3 \end{pmatrix}\)
- \(\text{adj}(BA) = \begin{pmatrix} 6 & -2 \\ -4 & 1 \end{pmatrix}\)
Comparing \(\text{adj}(AB)\) and \(\text{adj}(BA)\), we see that \(\begin{pmatrix} 4 & -2 \\ -7 & 3 \end{pmatrix} \neq \begin{pmatrix} 6 & -2 \\ -4 & 1 \end{pmatrix}\).
Therefore, Statement 2 is incorrect.
Statement 3: Adjugate Property Verification
This statement relates to the fundamental property of the adjugate matrix.
For any square matrix \(M\), we know that \(M \cdot \text{adj}(M) = |M| I_n\). Let \(M = AB\). Since \(A\) and \(B\) are non-singular, their product \(AB\) is also non-singular, meaning \(|AB| = |A||B| \neq 0\).
Applying the property to the matrix \(AB\), we get:
\((AB) \cdot \text{adj}(AB) = |AB| I_n\)Rearranging this equation, we obtain:
\((AB)\text{adj}(AB) - |AB|I_n = 0\)Here, \(0\) represents the null matrix (or zero matrix) of order \(n\). Thus, the expression \((AB)\text{adj}(AB) - |AB|I_n\) is indeed a null matrix.
Therefore, Statement 3 is correct.
Statements Correctness Summary
Evaluating the three provided statements regarding non-singular matrices \(A\) and \(B\):
- Statement 1: \(\text{adj}(AB) = (\text{adj}A)(\text{adj}B)\) - Incorrect.
- Statement 2: \(\text{adj}(AB) = \text{adj}(BA)\) - Incorrect.
- Statement 3: \((AB)\text{adj}(AB) - |AB|I_n\) is a null matrix - Correct.
Consequently, only one statement holds true.
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 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 :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?
- What is \(A(\text{adj } A)\) 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 ?
What is the value of the determinant of the inverse of the matrix \(\begin{bmatrix} -4 & -5 \\ 2 & 2 \end{bmatrix}\)?
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 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?
