Question

Which of the following statements is true? (1) A singular matrix has an inverse. (2) If a matrix doesn't have multiplicative inverse, it need not be a singular matrix. (3) If a, b are non-zero real numbers, then \(\left[\begin{array}{ll}a+b & a-b \\\ b-a & a+b\end{array}\right]\) is a non-singular matrix. (4) \(\left[\begin{array}{cc}5 & 2 \\ 3 & -1\end{array}\right]\) is a singular matrix.

Short answer

1. A singular matrix has an inverse. 2. If a matrix doesn't have a multiplicative inverse, it need not be a singular matrix. 3. If a, b are non-zero real numbers, then $\left[\begin{array}{ll}a+b & a-b \\\ b-a & a+b\end{array}\right]$ is a non-singular matrix. 4. \(\left[\begin{array}{cc}5 & 2 \\\ 3 & -1\end{array}\right]\) is a singular matrix. Answer: Statement 3 is true.

Step-by-step solution

(Statement 1)

(A singular matrix has an inverse.) A singular matrix has a determinant of zero, which means it does not have an inverse. If the determinant is non-zero, then the matrix has an inverse, and it is considered non-singular. So, statement (1) is false.

(Statement 2)

(If a matrix doesn't have a multiplicative inverse, it need not be a singular matrix.) If a matrix doesn't have a multiplicative inverse, it implies that its determinant is zero. And if a matrix has a determinant of zero, then it is a singular matrix. So, statement (2) is false.

(Statement 3)

(If a, b are non-zero real numbers, then $\left[\begin{array}{ll}a+b & a-b \\\ b-a & a+b\end{array}\right]$ is a non-singular matrix.) To check if this matrix is non-singular, we need to calculate its determinant. The determinant of a 2x2 matrix $\left[\begin{array}{ll}a & b \\\ c & d\end{array}\right]$ is given by: Det(A) = ad - bc For the given matrix, Det(A) = (a+b)(a+b) - (a-b)(b-a) = (a+b)^2 - (a^2 - b^2) = 4ab Since a and b are non-zero real numbers, 4ab will also be non-zero. Hence, the determinant is non-zero, which means the matrix is non-singular. So, statement (3) is true.

(Statement 4)

(\(\left[\begin{array}{cc}5 & 2 \\\ 3 & -1\end{array}\right]\) is a singular matrix.) To check if this matrix is singular, we need to calculate its determinant. Using the formula for the determinant of a 2x2 matrix: Det(A) = (5)(-1) - (2)(3) = -5 - 6 = -11 The determinant is non-zero, which means the matrix is non-singular. So, statement (4) is false. Based on our analysis, the true statement among the given options is (3).

Key concepts

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from its elements. This value is crucial in understanding the properties of the matrix, such as whether it has an inverse and the matrix's system-solving capacity (e.g., solving linear equations).

For a 2x2 matrix, the determinant can be calculated using the formula \(\text{Det}(A) = ad - bc\), where \(a, b, c, d\) are the elements of the matrix arranged as follows: \(\left[\begin{array}{cc} a & b \ c & d \end{array}\right]\). If the determinant is zero, the matrix is considered 'singular', meaning it cannot be inverted and represents a system of equations without a unique solution. Conversely, if the determinant is non-zero, the matrix is termed 'non-singular', and it has unique solutions for the associated linear equations.

Here is a simple bullet list to remember about the determinant:

• A non-zero determinant indicates a non-singular matrix.
• A zero determinant means the matrix is singular.
• The determinant helps determine the solvability of a system of linear equations.

Multiplicative Inverse of a Matrix

The multiplicative inverse of a matrix, often referred to as the inverse of a matrix, is analogous to the reciprocal of a number. Just as the reciprocal of a number \(a\) is \(1/a\), the inverse of a matrix \(A\), if it exists, is another matrix \(A^{-1}\) that when multiplied with \(A\), yields the identity matrix.

To qualify for an inverse, a matrix must be 'square' (same number of rows and columns) and 'non-singular'. Finding the inverse of a 2x2 matrix involves swapping some elements, changing the sign of others, and dividing by the determinant. The formula is \[A^{-1} = \frac{1}{\text{Det}(A)}\begin{bmatrix} d & -b \ -c & a \end{bmatrix}\] for a matrix \[A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\].

Remember the following points about matrix inverses:

• An inverse exists only for non-singular, square matrices.
• The product of a matrix and its inverse is the identity matrix.
• Computing the inverse requires a non-zero determinant.

Non-Singular Matrix

A non-singular matrix, sometimes referred to as an invertible or non-degenerate matrix, has some essential characteristics and implications in linear algebra. It is square, has a non-zero determinant, and it possesses a unique inverse.

Such matrices are of significant importance because they represent systems of linear equations that have a unique solution. Whether doing theoretical work or employing matrices in practical applications like computer graphics, understanding non-singular matrices is fundamental.

The following highlights summarize important aspects of non-singular matrices:

• They can be used to find unique solutions to linear equations.
• They have a non-zero determinant value.
• A matrix is non-singular if and only if it is invertible.

The exercise given earlier showcased a practical example involving the determinant to confirm the non-singularity of the matrix formed by real numbers \(a\) and \(b\).