Determine The Inverse Of The Matrix $ C = \begin{bmatrix} 6 & -7 \ -8 & 9 \end{bmatrix} }$Choose The Correct Inverse From The Options Below A. ${ C^{-1 = \begin{bmatrix} 9 & 7 \ 8 & 6 \end{bmatrix} }$B. $[ C^{-1} =

Introduction

In linear algebra, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. One of the fundamental operations in matrix algebra is finding the inverse of a matrix. The inverse of a matrix is a special matrix that, when multiplied by the original matrix, results in the identity matrix. In this article, we will determine the inverse of the matrix ${ C = \begin{bmatrix} 6 & -7 \ -8 & 9 \end{bmatrix} }$ and choose the correct inverse from the given options.

What is a Matrix Inverse?

A matrix inverse is a matrix that, when multiplied by the original matrix, results in the identity matrix. The identity matrix is a special matrix that has 1s on the main diagonal and 0s elsewhere. The inverse of a matrix is denoted by the symbol $^{-1}$.

Properties of Matrix Inverse

The matrix inverse has several important properties:

• The product of a matrix and its inverse is the identity matrix.
• The product of the inverse of a matrix and the original matrix is the identity matrix.
• The inverse of a matrix is unique.

Determining the Inverse of a Matrix

To determine the inverse of a matrix, we can use the following formula:

$$C^{-1} = \frac{1}{\det(C)} \begin{bmatrix} c_{22} & -c_{12} \\ -c_{21} & c_{11} \end{bmatrix}$$

where $\det(C)$ is the determinant of the matrix $C$, and $c_{ij}$ are the elements of the matrix $C$.

Calculating the Determinant of the Matrix

The determinant of a 2x2 matrix is calculated as follows:

$$\det(C) = c_{11}c_{22} - c_{12}c_{21}$$

In this case, the determinant of the matrix $C$ is:

$$\det(C) = 6(9) - (-7)(-8) = 54 - 56 = -2$$

Calculating the Inverse of the Matrix

Now that we have the determinant of the matrix, we can calculate the inverse of the matrix using the formula:

$$C^{-1} = \frac{1}{\det(C)} \begin{bmatrix} c_{22} & -c_{12} \\ -c_{21} & c_{11} \end{bmatrix}$$

Substituting the values, we get:

$$C^{-1} = \frac{1}{-2} \begin{bmatrix} 9 & 7 \\ 8 & 6 \end{bmatrix}$$

Simplifying, we get:

$$C^{-1} = \begin{bmatrix} -\frac{9}{2} & -\frac{7}{2} \\ -4 & -3 \end{bmatrix}$$

Choosing the Correct Inverse

Now that we have calculated the inverse of the matrix, we can choose the correct inverse from the given options.

Option A: ${ C^{-1} = \begin{bmatrix} 9 & 7 \ 8 & 6 \end{bmatrix} }$

Option B: ${ C^{-1} = \begin{bmatrix} -\frac{9}{2} & -\frac{7}{2} \ -4 & -3 \end{bmatrix} }$

Based on our calculation, the correct inverse of the matrix $C$ is:

$$C^{-1} = \begin{bmatrix} -\frac{9}{2} & -\frac{7}{2} \\ -4 & -3 \end{bmatrix}$$

Therefore, the correct answer is:

The final answer is B.

Conclusion

In this article, we determined the inverse of the matrix ${ C = \begin{bmatrix} 6 & -7 \ -8 & 9 \end{bmatrix} }$ using the formula for the inverse of a matrix. We calculated the determinant of the matrix and then used the formula to calculate the inverse of the matrix. Finally, we chose the correct inverse from the given options. The correct answer is B.

References

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

Further Reading

• [1] Inverse of a Matrix, Wikipedia
• [2] Determinant of a Matrix, Wikipedia

Mathematics Discussion Forum

Introduction

In our previous article, we determined the inverse of the matrix ${ C = \begin{bmatrix} 6 & -7 \ -8 & 9 \end{bmatrix} }$ using the formula for the inverse of a matrix. In this article, we will answer some frequently asked questions about determining the inverse of a matrix.

Q: What is the formula for the inverse of a matrix?

A: The formula for the inverse of a matrix is:

$$C^{-1} = \frac{1}{\det(C)} \begin{bmatrix} c_{22} & -c_{12} \\ -c_{21} & c_{11} \end{bmatrix}$$

where $\det(C)$ is the determinant of the matrix $C$, and $c_{ij}$ are the elements of the matrix $C$.

Q: How do I calculate the determinant of a matrix?

A: The determinant of a 2x2 matrix is calculated as follows:

$$\det(C) = c_{11}c_{22} - c_{12}c_{21}$$

Q: What is the significance of the determinant in determining the inverse of a matrix?

A: The determinant is used to calculate the inverse of a matrix. If the determinant is zero, then the matrix is singular and does not have an inverse.

Q: Can a matrix have multiple inverses?

A: No, a matrix can only have one inverse.

Q: How do I choose the correct inverse from the given options?

A: To choose the correct inverse, you need to calculate the inverse of the matrix using the formula and then compare it with the given options.

Q: What are some common mistakes to avoid when determining the inverse of a matrix?

A: Some common mistakes to avoid when determining the inverse of a matrix include:

• Not calculating the determinant correctly
• Not using the correct formula for the inverse of a matrix
• Not checking if the matrix is singular before calculating the inverse

Q: Can you provide an example of how to determine the inverse of a matrix?

A: Let's consider the matrix ${ C = \begin{bmatrix} 2 & 3 \ 4 & 5 \end{bmatrix} }$. To determine the inverse of this matrix, we need to calculate the determinant and then use the formula for the inverse of a matrix.

The determinant of the matrix is:

$$\det(C) = 2(5) - 3(4) = 10 - 12 = -2$$

Now, we can calculate the inverse of the matrix using the formula:

$$C^{-1} = \frac{1}{\det(C)} \begin{bmatrix} c_{22} & -c_{12} \\ -c_{21} & c_{11} \end{bmatrix}$$

Substituting the values, we get:

$$C^{-1} = \frac{1}{-2} \begin{bmatrix} 5 & -3 \\ -4 & 2 \end{bmatrix}$$

Simplifying, we get:

$$C^{-1} = \begin{bmatrix} -\frac{5}{2} & \frac{3}{2} \\ 2 & -1 \end{bmatrix}$$

Therefore, the inverse of the matrix ${ C = \begin{bmatrix} 2 & 3 \ 4 & 5 \end{bmatrix} }$ is:

$$C^{-1} = \begin{bmatrix} -\frac{5}{2} & \frac{3}{2} \\ 2 & -1 \end{bmatrix}$$

Conclusion

In this article, we answered some frequently asked questions about determining the inverse of a matrix. We provided examples and explanations to help you understand the concept better. If you have any more questions or need further clarification, please feel free to ask.

References

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

Further Reading

• [1] Inverse of a Matrix, Wikipedia
• [2] Determinant of a Matrix, Wikipedia

Mathematics Discussion Forum

If you have any questions or comments about this article, please feel free to post them in the mathematics discussion forum.