Which Matrix Is The Inverse Of $\left[\begin{array}{cc}-2 & 5 \ 1 & -2\end{array}\right]?A. $\left[\begin{array}{cc}2 & -5 \ -1 & 2\end{array}\right]B. $\left[\begin{array}{cc}-2 & -5 \ 1 & 2\end{array}\right]C.

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In linear algebra, the inverse of a matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix. In this article, we will explore how to find the inverse of a 2x2 matrix and apply this concept to a specific problem.

What is a 2x2 Matrix?

A 2x2 matrix is a square matrix with two rows and two columns. It has the form:

[abcd]\left[\begin{array}{cc}a & b \\ c & d\end{array}\right]

where a, b, c, and d are numbers.

The Formula for the Inverse of a 2x2 Matrix

The formula for the inverse of a 2x2 matrix is:

1ad−bc[d−b−ca]\frac{1}{ad - bc} \left[\begin{array}{cc}d & -b \\ -c & a\end{array}\right]

where ad - bc is the determinant of the matrix.

Finding the Inverse of a Specific Matrix

Now, let's apply this formula to the matrix:

[−251−2]\left[\begin{array}{cc}-2 & 5 \\ 1 & -2\end{array}\right]

First, we need to calculate the determinant of the matrix:

ad−bc=(−2)(−2)−(5)(1)=4−5=−1ad - bc = (-2)(-2) - (5)(1) = 4 - 5 = -1

Next, we plug the values into the formula:

1−1[−2−512]\frac{1}{-1} \left[\begin{array}{cc}-2 & -5 \\ 1 & 2\end{array}\right]

Simplifying the expression, we get:

[25−1−2]\left[\begin{array}{cc}2 & 5 \\ -1 & -2\end{array}\right]

Which Matrix is the Inverse?

Now, let's compare the result with the answer choices:

A. [2−5−12]\left[\begin{array}{cc}2 & -5 \\ -1 & 2\end{array}\right] B. [−2−512]\left[\begin{array}{cc}-2 & -5 \\ 1 & 2\end{array}\right] C. [25−1−2]\left[\begin{array}{cc}2 & 5 \\ -1 & -2\end{array}\right]

The correct answer is C. [25−1−2]\left[\begin{array}{cc}2 & 5 \\ -1 & -2\end{array}\right].

Conclusion

In this article, we learned how to find the inverse of a 2x2 matrix using the formula:

1ad−bc[d−b−ca]\frac{1}{ad - bc} \left[\begin{array}{cc}d & -b \\ -c & a\end{array}\right]

We applied this formula to a specific matrix and found the inverse. We also compared the result with the answer choices and determined that the correct answer is C. [25−1−2]\left[\begin{array}{cc}2 & 5 \\ -1 & -2\end{array}\right].

Frequently Asked Questions

  • What is the inverse of a matrix?
  • How do I find the inverse of a 2x2 matrix?
  • What is the formula for the inverse of a 2x2 matrix?

Answer

  • The inverse of a matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix.
  • To find the inverse of a 2x2 matrix, you need to calculate the determinant and plug the values into the formula.
  • The formula for the inverse of a 2x2 matrix is:

1ad−bc[d−b−ca]\frac{1}{ad - bc} \left[\begin{array}{cc}d & -b \\ -c & a\end{array}\right]

References

  • Linear Algebra by Gilbert Strang
  • Matrix Algebra by James E. Gentle
  • Inverse of a Matrix by Wolfram MathWorld
    Inverse of a Matrix: Frequently Asked Questions =====================================================

In the previous article, we discussed how to find the inverse of a 2x2 matrix using the formula:

1ad−bc[d−b−ca]\frac{1}{ad - bc} \left[\begin{array}{cc}d & -b \\ -c & a\end{array}\right]

However, we know that there are many more questions and doubts that readers may have. In this article, we will address some of the most frequently asked questions about the inverse of a matrix.

Q: What is the inverse of a matrix?

A: The inverse of a matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix. In other words, if we have a matrix A and its inverse A^-1, then:

AA^-1 = A^-1A = I

where I is the identity matrix.

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

A: To find the inverse of a 2x2 matrix, you need to calculate the determinant and plug the values into the formula:

1ad−bc[d−b−ca]\frac{1}{ad - bc} \left[\begin{array}{cc}d & -b \\ -c & a\end{array}\right]

where ad - bc is the determinant of the matrix.

Q: What is the formula for the inverse of a 2x2 matrix?

A: The formula for the inverse of a 2x2 matrix is:

1ad−bc[d−b−ca]\frac{1}{ad - bc} \left[\begin{array}{cc}d & -b \\ -c & a\end{array}\right]

Q: What is the determinant of a matrix?

A: The determinant of a matrix is a scalar value that can be used to determine the invertibility of the matrix. For a 2x2 matrix, the determinant is calculated as:

ad - bc

where a, b, c, and d are the elements of the matrix.

Q: How do I know if a matrix is invertible?

A: A matrix is invertible if its determinant is non-zero. In other words, if the determinant is zero, then the matrix is not invertible.

Q: What is the inverse of a matrix with a zero determinant?

A: If a matrix has a zero determinant, then it is not invertible. In this case, the inverse of the matrix does not exist.

Q: Can I find the inverse of a matrix using a calculator?

A: Yes, you can find the inverse of a matrix using a calculator. Most calculators have a built-in function to calculate the inverse of a matrix.

Q: How do I use a calculator to find the inverse of a matrix?

A: To use a calculator to find the inverse of a matrix, follow these steps:

  1. Enter the matrix into the calculator.
  2. Press the "inv" or "inverse" button to calculate the inverse of the matrix.
  3. The calculator will display the inverse of the matrix.

Q: What are some common applications of the inverse of a matrix?

A: The inverse of a matrix has many applications in mathematics and science, including:

  • Solving systems of linear equations
  • Finding the solution to a linear system
  • Calculating the determinant of a matrix
  • Finding the inverse of a matrix

Conclusion

In this article, we addressed some of the most frequently asked questions about the inverse of a matrix. We discussed how to find the inverse of a 2x2 matrix, what the formula for the inverse of a 2x2 matrix is, and how to use a calculator to find the inverse of a matrix. We also covered some common applications of the inverse of a matrix.

Frequently Asked Questions

  • What is the inverse of a matrix?
  • How do I find the inverse of a 2x2 matrix?
  • What is the formula for the inverse of a 2x2 matrix?
  • What is the determinant of a matrix?
  • How do I know if a matrix is invertible?
  • What is the inverse of a matrix with a zero determinant?
  • Can I find the inverse of a matrix using a calculator?
  • How do I use a calculator to find the inverse of a matrix?
  • What are some common applications of the inverse of a matrix?

Answer

  • The inverse of a matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix.
  • To find the inverse of a 2x2 matrix, you need to calculate the determinant and plug the values into the formula.
  • The formula for the inverse of a 2x2 matrix is:

1ad−bc[d−b−ca]\frac{1}{ad - bc} \left[\begin{array}{cc}d & -b \\ -c & a\end{array}\right]

  • The determinant of a matrix is a scalar value that can be used to determine the invertibility of the matrix.
  • A matrix is invertible if its determinant is non-zero.
  • If a matrix has a zero determinant, then it is not invertible.
  • Yes, you can find the inverse of a matrix using a calculator.
  • To use a calculator to find the inverse of a matrix, follow these steps: Enter the matrix into the calculator, press the "inv" or "inverse" button to calculate the inverse of the matrix, and the calculator will display the inverse of the matrix.
  • The inverse of a matrix has many applications in mathematics and science, including solving systems of linear equations, finding the solution to a linear system, calculating the determinant of a matrix, and finding the inverse of a matrix.