Let A and B be two \(3 × 3\) non-zero real matrices such that AB is a zero matrix. Then
- the system of linear equations AX = 0 has a unique solution
- the system of linear equations AX = 0 has infinitely many solutions
- B is an invertible matrix
- adj(A) is an invertible matrix
The Correct Option is B
Approach Solution - 1
To understand why the given statement "the system of linear equations AX = 0 has infinitely many solutions" is correct, let's analyze the scenario where \( A \) and \( B \) are non-zero \( 3 \times 3 \) matrices such that their product is a zero matrix, i.e., \( AB = 0 \).
1. Given that \( AB = 0 \) (the zero matrix), this implies that the rank of the matrix \( A \) must be less than 3. Otherwise, if \( A \) were invertible (rank 3), the product would not result in a zero matrix unless \( B \) was also a zero matrix, which contradicts the given non-zero condition.
2. If the rank of \( A \) is less than 3 (say less than full rank), it means that \( A \) does not have full row or column rank. Therefore, the homogeneous system of equations \( AX = 0 \) must have infinitely many solutions.
3. By linear algebra, a homogeneous system \( AX = 0 \) has:
- A unique solution (only the trivial solution) if and only if \( A \) is full rank (invertible).
- Infinitely many solutions if \( A \) is not full rank (i.e., rank is less than the number of variables).
Since \( AB = 0 \) and both \( A \) and \( B \) are non-zero, \( A \) cannot be of full rank, hence the system \( AX = 0 \) has infinitely many solutions.
4. Analyzing the other options:
- The system of linear equations AX = 0 has a unique solution: As explained, this is incorrect because \( A \) is not full rank.
- B is an invertible matrix: If \( B \) were invertible, multiplying both sides of \( AB = 0 \) by \( B^{-1} \) would give \( A = 0 \), contradicting the non-zero condition of \( A \).
- adj(A) is an invertible matrix: The adjugate matrix adj(A) is invertible if and only if \( A \) is not singular, which would imply \( A \) has full rank, which we've determined is not the case.
Thus, the correct conclusion is that the system of linear equations \( AX = 0 \) has infinitely many solutions, due to \( A \) being a non-full rank matrix.
Approach Solution -2
AB is zero matrix
\(⇒ |A| = |B| = 0\)
Hence, neither A nor B is invertible
If \(|A| = 0\)
\(⇒ |adj A| = 0\) so adj A is not invertible
\(AX = 0\) is homogeneous system and \(|A| = 0\)
Therefore, it is having infinitely many solutions.
So, the correct option is (B): the system of linear equations \(AX = 0\) has infinitely many solutions
Top JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
Top JEE Main Matrices Questions
Let \[ R = \begin{pmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{pmatrix} \text{ be a non-zero } 3 \times 3 \text{ matrix, where} \]
\[ x = \sin \theta, \quad y = \sin \left( \theta + \frac{2\pi}{3} \right), \quad z = \sin \left( \theta + \frac{4\pi}{3} \right) \]
and \( \theta \neq 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi \). For a square matrix \( M \), let \( \text{trace}(M) \) denote the sum of all the diagonal entries of \( M \). Then, among the statements:
- \(\text{Trace}(R) = 0\)
- If \(\text{trace}(\text{adj}(\text{adj}(R))) = 0\), then \( R \) has exactly one non-zero entry.
Which of the following is true?
- Let \( A \) be a \( 2 \times 2 \) real matrix and \( I \) be the identity matrix of order 2. If the roots of the equation \[ |A - xI| = 0 \] be \( -1 \) and \( 3 \), then the sum of the diagonal elements of the matrix \( A^2 \) is .....
- Consider the matrix \( f(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix} \).
Given below are two statements:
Statement I: \( f(-x) \) is the inverse of the matrix \( f(x) \).
Statement II: \( f(x) f(y) = f(x + y) \).
In the light of the above statements, choose the correct answer from the options given below:" - Let \( A = \begin{bmatrix} 2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix} \), \( B = [B_1, B_2, B_3] \), where \( B_1, B_2, B_3 \) are column matrices, and \( AB_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \), \( AB_2 = \begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix} \), \( AB_3 = \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix} \).
If \(\alpha = |B|\) and \(\beta\) is the sum of all the diagonal elements of B, then \(\alpha^3 + \beta^3\) is equal to _____. - If \( A = \begin{bmatrix} \sqrt{2} & 1 \\ -1 & \sqrt{2} \end{bmatrix} \), \( B = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} \), \( C = ABA^\top \) and \( X = A^\top C^2 A \), then \( \det (X) \) is equal to:
Top JEE Main Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
Concepts Used:
Matrices
Matrix:
A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix is determined by the number of rows and columns in the matrix.
The basic operations that can be performed on matrices are:
- Addition of Matrices - The addition of matrices addition can only be possible if the number of rows and columns of both the matrices are the same.
- Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
- Scalar Multiplication - The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, is called scalar multiplication.
- Multiplication of Matrices - Matrices multiplication is defined only if the number of columns in the first matrix and rows in the second matrix are equal.
- Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.