(b) Given That The Determinant Of The Matrix A = ( 2 X − 1 1 4 − 2 ) A=\begin{pmatrix} 2x-1 & 1 \\ 4 & -2 \end{pmatrix} A = ( 2 X − 1 4 ​ 1 − 2 ​ ) Is -10, Find:(i) The Value Of X X X .(ii) The Inverse Of Matrix A A A .

Introduction

In linear algebra, the determinant of a matrix is a scalar value that can be used to describe the scaling effect of the matrix on a region of space. It is also used to determine the invertibility of a matrix. In this article, we will discuss how to find the value of $x$ given that the determinant of the matrix $A$ is -10, and then find the inverse of matrix $A$.

The Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix $A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is given by the formula:

$$\det(A) = ad - bc$$

In this case, the matrix $A$ is given by:

$$A=\begin{pmatrix} 2x-1 & 1 \\ 4 & -2 \end{pmatrix}$$

We are given that the determinant of matrix $A$ is -10, so we can set up the equation:

$$(2x-1)(-2) - (1)(4) = -10$$

Solving for $x$

To solve for $x$, we can start by simplifying the equation:

$$-4x + 2 - 4 = -10$$

Combine like terms:

$$-4x - 2 = -10$$

Add 2 to both sides:

$$-4x = -8$$

Divide both sides by -4:

$$x = 2$$

Therefore, the value of $x$ is 2.

The Inverse of a 2x2 Matrix

The inverse of a 2x2 matrix $A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is given by the formula:

$$A^{-1} = \frac{1}{\det(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$

In this case, the determinant of matrix $A$ is -10, so we can plug in the values:

$$A^{-1} = \frac{1}{-10} \begin{pmatrix} -2 & -1 \\ -4 & 2x-1 \end{pmatrix}$$

Simplify the expression:

$$A^{-1} = \begin{pmatrix} \frac{1}{5} & \frac{1}{10} \\ \frac{2}{5} & \frac{2x-1}{10} \end{pmatrix}$$

Now that we have found the value of $x$, we can substitute it into the expression for the inverse:

$$A^{-1} = \begin{pmatrix} \frac{1}{5} & \frac{1}{10} \\ \frac{2}{5} & \frac{4-1}{10} \end{pmatrix}$$

Simplify the expression:

$$A^{-1} = \begin{pmatrix} \frac{1}{5} & \frac{1}{10} \\ \frac{2}{5} & \frac{3}{10} \end{pmatrix}$$

Therefore, the inverse of matrix $A$ is:

$$A^{-1} = \begin{pmatrix} \frac{1}{5} & \frac{1}{10} \\ \frac{2}{5} &frac{3}{10} \end{pmatrix}$$

Conclusion

In this article, we have discussed how to find the value of $x$ given that the determinant of the matrix $A$ is -10, and then find the inverse of matrix $A$. We have used the formula for the determinant of a 2x2 matrix and the formula for the inverse of a 2x2 matrix to solve for $x$ and find the inverse of matrix $A$. The value of $x$ is 2, and the inverse of matrix $A$ is:

$$A^{-1} = \begin{pmatrix} \frac{1}{5} & \frac{1}{10} \\ \frac{2}{5} & \frac{3}{10} \end{pmatrix}$$

References

• [1] Linear Algebra and Its Applications, 4th Edition, by Gilbert Strang
• [2] Matrix Algebra, 2nd Edition, by James E. Gentle
  Determinant of a Matrix and Its Inverse: Q&A =====================================================

Introduction

In our previous article, we discussed how to find the value of $x$ given that the determinant of the matrix $A$ is -10, and then find the inverse of matrix $A$. In this article, we will answer some frequently asked questions related to the determinant of a matrix and its inverse.

Q&A

Q: What is the determinant of a matrix?

A: The determinant of a matrix is a scalar value that can be used to describe the scaling effect of the matrix on a region of space. It is also used to determine the invertibility of a matrix.

Q: How do I find the determinant of a 2x2 matrix?

A: The determinant of a 2x2 matrix $A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is given by the formula:

$$\det(A) = ad - bc$$

Q: What is the inverse of a matrix?

A: The inverse of a matrix $A$ is a matrix $A^{-1}$ such that $AA^{-1} = A^{-1}A = I$, where $I$ is the identity matrix.

Q: How do I find the inverse of a 2x2 matrix?

A: The inverse of a 2x2 matrix $A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is given by the formula:

$$A^{-1} = \frac{1}{\det(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$

Q: What is the relationship between the determinant and the inverse of a matrix?

A: The determinant of a matrix is used to determine the invertibility of the matrix. If the determinant of a matrix is non-zero, then the matrix is invertible. The inverse of a matrix is given by the formula:

$$A^{-1} = \frac{1}{\det(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$

Q: Can you give an example of finding the inverse of a matrix?

A: Let's consider the matrix $A=\begin{pmatrix} 2x-1 & 1 \\ 4 & -2 \end{pmatrix}$. We are given that the determinant of matrix $A$ is -10. We can find the inverse of matrix $A$ using the formula:

$$A^{-1} = \frac{1}{-10} \begin{pmatrix} -2 & -1 \\ -4 & 2x-1 \end{pmatrix}$$

Simplify the expression:

$$A^{-1} = \begin{pmatrix} \frac{1}{5} & \frac{1}{10} \\ \frac{2}{5} & \frac{2x-1}{10} \end{pmatrix}$$

Now that we have found the value of $x$, we can substitute it into the expression for the inverse:

$$A^{-1} = \begin{pmatrix} \frac{1}{5} & \frac{1}{10} \\ \frac{2}{5} & \frac{4-1}{10} \end{pmatrix}$$

Simplify the expression:

A^{-1} = \begin{pmatrix} \frac1}{5} & \frac{1}{10} \\ \frac{2}{5} & \frac{3}{10} \end{pmatrix}

Therefore, the inverse of matrix $A$ is:

$$A^{-1} = \begin{pmatrix} \frac{1}{5} & \frac{1}{10} \\ \frac{2}{5} & \frac{3}{10} \end{pmatrix}$$

Conclusion

In this article, we have answered some frequently asked questions related to the determinant of a matrix and its inverse. We have discussed the formula for the determinant of a 2x2 matrix, the formula for the inverse of a 2x2 matrix, and the relationship between the determinant and the inverse of a matrix. We have also given an example of finding the inverse of a matrix.

References

• [1] Linear Algebra and Its Applications, 4th Edition, by Gilbert Strang
• [2] Matrix Algebra, 2nd Edition, by James E. Gentle