Question

A matrix that has no inverse is called a(n): (a) zero matrix (b) nonsingular matrix (c) identity matrix (d) singular matrix

Short answer

The correct answer is (d) singular matrix.

Step-by-step solution

Understand the definition

A matrix with no inverse is one for which there does not exist another matrix that can be multiplied with it to produce the identity matrix. This type of matrix is crucial in linear algebra.

Review the Options

Look at the given options:(a) Zero matrix(b) Nonsingular matrix(c) Identity matrix(d) Singular matrix

Evaluate each Option

Evaluate each term to see if it matches the definition of a matrix with no inverse:(a) Zero matrix: This is a matrix with all elements being zero; it indeed has no inverse.(b) Nonsingular matrix: This is a matrix that has an inverse. This is not the correct answer.(c) Identity matrix: This is a special type of matrix that acts as a multiplicative identity, and it does have an inverse.(d) Singular matrix: This is a matrix that has no inverse because its determinant is zero.

Choose the Correct Answer

From the evaluation above, both zero matrix and singular matrix do not have inverses, but the term commonly used in linear algebra to describe a matrix with no inverse is a 'singular matrix'. Therefore, the correct answer is (d) singular matrix.

Key concepts

Determinant

The determinant is a special number that can be calculated from a square matrix. It plays a critical role in understanding whether a matrix is invertible or not. To calculate the determinant of a 2x2 matrix \[ A = \begin{pmatrix} a & b \ c & d \ \ \ \ \ \ \ \ \end{pmatrix} \], you use the formula: \[ det(A) = ad - bc \].

Identity Matrix

An identity matrix is a special type of square matrix with ones on the diagonal and zeros elsewhere. It acts as the multiplicative identity in matrix operations. This means any matrix multiplied by the identity matrix remains unchanged.