Linear Algebra - Free MCQ Practice Test with solutions, GATE CSE (CSE)

A singular matrix is defined as one that does not have an inverse, which occurs when its determinant is zero. This implies that at least one of its eigenvalues must also be zero. Therefore, option B is correct.
In contrast, a non-singular matrix has a non-zero determinant and all eigenvalues are non-zero.

The trace of a matrix is defined as the sum of its diagonal elements, which is also equal to the sum of its eigenvalues. Therefore, option B is correct.
This relationship holds true regardless of whether the eigenvalues are positive, negative, or repeated.

Option A is correct because the determinant equals the product of the eigenvalues.

The determinant of a square matrix is the product of its eigenvalues (counted with algebraic multiplicity).

Compute the product: 2 × (-3) × 4 = -24.

Therefore, the determinant is -24, so Option A is the correct choice.

An orthogonal matrix A satisfies the property A^TA = I, where I is the identity matrix. Therefore, option A is correct.
This property implies that the columns (and rows) of A are orthonormal vectors.

The trace of a matrix is the sum of its eigenvalues. Thus, the trace of matrix A is 5 + 3 = 8. Therefore, the correct answer is option A.
This relationship is fundamental in linear algebra and helps in characterizing the matrix.

A symmetric matrix has the property that all its eigenvalues are real numbers. Therefore, option B is correct.
This is a significant property that allows for real solutions in various applications.

If the rows of a matrix are linearly dependent, then the determinant of that matrix is zero. Thus, option C is correct.
This indicates that the matrix does not span the full space, leading to a loss of dimensionality.

The characteristic polynomial of a matrix A, which has eigenvalues λ₁, λ₂, and λ₃, is given by the expression (λ - λ₁)(λ - λ₂)(λ - λ₃). Therefore, option A is correct.
This polynomial is crucial for finding eigenvalues and understanding the behavior of the matrix.

Interchanging two rows of a matrix changes the sign of its determinant. Thus, option B is correct.
This property is important in various matrix operations and transformations.

A one-to-one transformation means that each input corresponds to a unique output, meaning no two different inputs map to the same output. Thus, option B is correct.
This characteristic is vital for the invertibility of the transformation.

Adding a multiple of one row to another does not change the determinant of the matrix. Hence, option A is correct.
This operation is commonly used in Gaussian elimination without affecting the determinant's value.

The rank of a matrix is the number of non-zero eigenvalues. Here, the non-zero eigenvalues are 2 and 3. Therefore, the rank of A is 2, making option C correct.
This highlights the relationship between eigenvalues and the linear independence of the matrix columns.

When a matrix is multiplied by a scalar k, all eigenvalues of that matrix are also multiplied by k. Thus, option A is correct.
This property demonstrates how scalar multiplication affects the transformation represented by the matrix.

The rank of a matrix is defined as the maximum number of linearly independent row or column vectors. It cannot exceed the smaller of the number of rows or columns. Therefore, option C is correct.
This is a fundamental concept in linear algebra and essential for understanding matrix properties.

If the determinant of a square coefficient matrix is non-zero, it indicates that the matrix is invertible, leading to a unique solution for the system of equations. Thus, option C is correct.
This property is crucial for solving linear equations efficiently.