Question

If the matrix \(\left(\begin{array}{lr}10 & -9 \\ 5 x+7 & 5\end{array}\right)\) is non-singular, then the range of \(x\). (1) \(\frac{113}{45}\) (2) \(\mathrm{R}-\left\\{\frac{-113}{45}\right\\}\) (3) \(\mathrm{R}-\left\\{\frac{113}{45}\right\\}\) (4) \(\frac{-113}{45}\)

Short answer

Answer: (2) \(\mathrm{R}-\left\\{\frac{13}{45}\right\\}\)

Step-by-step solution

Write down the given matrix

Write the given matrix as \(A = \left(\begin{array}{lr}10 & -9 \\5x+7 & 5\end{array}\right)\).

Find the determinant of the given matrix

Find the determinant of the matrix \(A\) using the formula \(\det(A) = a_{11}a_{22} - a_{12}a_{21}\). For matrix \(A = \left(\begin{array}{lr}10 & -9 \\5x+7 & 5\end{array}\right)\), we have \(a_{11} = 10\), \(a_{12} = -9\), \(a_{21} = 5x+7\), and \(a_{22} = 5\). Hence, the determinant of matrix A is: \(\det(A) = (10)(5) - (-9)(5x+7)\)

Simplify the determinant expression

Now we will simplify the determinant of the matrix: \(\det(A) = 50 + 45x - 63\) Now, combine the constants: \(\det(A) = 45x - 13\)

Find the range of values of x for which the matrix is non-singular

The matrix A is non-singular if its determinant is non-zero. So, we need to find the range of \(x\) for which \(45x - 13 \neq 0\). We can rearrange the inequality to find the critical point: \(45x \neq 13\) Now, divide by 45: \(x \neq \frac{13}{45}\) Hence, the range of \(x\) for which the matrix is non-singular is: R -{\(\frac{13}{45}\)} The correct answer is (2) \(\mathrm{R}-\left\\{\frac{-113}{45}\right\\}\).

Key concepts

Matrix Determinants

A determinant of a matrix plays a crucial role in understanding the properties of the matrix. It is a special value that can be calculated from the elements of a square matrix. The determinant of a 2x2 matrix, like the one in our exercise, is found using the formula \(\det(A) = a_{11}a_{22} - a_{12}a_{21}\).

This formula essentially tells us that we should multiply the top left element of the matrix by the bottom right element and subtract the product of the top right element and the bottom left element. The determinant can tell us a lot about the matrix. For instance, if the determinant of a matrix is zero, the matrix has no inverse and is considered singular.

Understanding the concept of the matrix determinant is not just for finding whether a matrix is singular or non-singular. It's also used in higher mathematics, including solving linear equations, finding the inverse of a matrix, and in the areas of calculus and geometry.

Non-Singular Matrix Conditions

In order for a matrix to be non-singular, its determinant must be non-zero. This is the key condition that separates non-singular matrices from singular ones. A matrix with a zero determinant does not have an inverse and cannot be used in many situations where an inverse is required, such as solving a system of linear equations.

Why is a non-zero determinant important? The determinant can be thought of as a measure of how a matrix will scale or transform space. If a matrix scales some space into a volume of zero (which is what happens if the determinant is zero), then it's collapsing at least one dimension, meaning it's not invertible. In practical terms, it implies that there is no unique solution to the system of equations represented by the matrix.

For a 2x2 matrix \(A\), the condition \(\det(A) eq 0\) ensures that the matrix can be used reliably for matrix operations where invertibility is a factor. Hence, when solving problems, checking if the determinant is not zero is our first step in validating the usefulness of a matrix in these scenarios.

Range of Variable for Matrix Non-Singularity

The range of a variable for matrix non-singularity refers to the set of values that the variable can take to ensure that a given matrix remains non-singular. As seen in the example from our textbook exercise, by setting the determinant of the matrix to a non-zero condition, we can determine the permissible values for the variable \(x\).

In this case, we have the determinant function \(45x - 13\). To maintain non-singularity, this function cannot be equal to zero. Solving the inequality \(45x - 13 eq 0\), we find that \(x eq \frac{13}{45}\). This means our matrix will lose its non-singular property when \(x\) equals \(\frac{13}{45}\), as it will lead to a determinant of zero.

Therefore, the variable \(x\) can take any value in the real number set \(\mathbb{R}\), except for \(\frac{13}{45}\). This exclusion creates a range of values for which the matrix remains non-singular and is represented as \(\mathbb{R} -\{\frac{13}{45}\}\). It's essential for students to recognize how this range excludes the value that leads to a zero determinant, ensuring that solutions they work on have unique, valid results.