Step 3: Multiply By The Scalar (1 Over The Determinant):$\[ A^{-1}=\begin{bmatrix} a_{11} & A_{12} \\ a_{21} & A_{22} \end{bmatrix} \\]$\[ a_{11}=\square \quad A_{12}=\square \\]$\[ a_{22}=\square \quad A_{21}=\square \\]

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Introduction

In the previous steps, we have learned how to find the inverse of a 2x2 matrix. We have also learned how to calculate the determinant of a 2x2 matrix. Now, in this step, we will learn how to multiply the matrix obtained in the previous step by the scalar (1 over the determinant). This is the final step in finding the inverse of a 2x2 matrix.

What is a Scalar?

A scalar is a number that is used to multiply a matrix. In this case, the scalar is (1 over the determinant). The determinant of a 2x2 matrix is calculated as follows:

Determinant of a 2x2 Matrix

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

∣abcd∣=ad−bc\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc

Calculating the Determinant

Let's calculate the determinant of the matrix obtained in the previous step.

A=[a11a12a21a22]A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}

The determinant of A is calculated as follows:

∣a11a12a21a22∣=a11a22−a12a21\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} = a_{11}a_{22} - a_{12}a_{21}

Multiplying by the Scalar

Now that we have calculated the determinant, we can multiply the matrix obtained in the previous step by the scalar (1 over the determinant).

A−1=1∣a11a12a21a22∣[a11a12a21a22]A^{-1} = \frac{1}{\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}} \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}

A−1=1a11a22−a12a21[a11a12a21a22]A^{-1} = \frac{1}{a_{11}a_{22} - a_{12}a_{21}} \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}

Simplifying the Expression

We can simplify the expression by multiplying the numerator and denominator by the conjugate of the denominator.

A−1=1a11a22−a12a21[a11a12a21a22]A^{-1} = \frac{1}{a_{11}a_{22} - a_{12}a_{21}} \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}

A−1=1(a11a22−a12a21)[a11a12a21a22]A^{-1} = \frac{1}{(a_{11}a_{22} - a_{12}a_{21})} \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}

Final Answer

The final answer is:

A−1=[a22a11a22−a12a21−a12a11a22−a12a21−a21a11a22−a12a21a11a11a22−a12a21]A^{-1} = \begin{bmatrix} \frac{a_{22}}{a_{11}a_{22} - a_{12}a_{21}} & \frac{-a_{12}}{a_{11}a_{22} - a_{12}a_{21}} \\ \frac{-a_{21}}{a_{11}a_{22} - a_{12}a_{21}} & \frac{a_{11}}{a_{11}a_{22} - a_{12}a_{21}} \end{bmatrix}

Conclusion

In this step, we have learned how to multiply the matrix obtained in the previous step by the scalar (1 over the determinant). This is the final step in finding the inverse of a 2x2 matrix. We have also learned how to calculate the determinant of a 2x2 matrix and how to simplify the expression by multiplying the numerator and denominator by the conjugate of the denominator.

Example

Let's consider an example to illustrate the concept.

Suppose we have the following matrix:

A=[2345]A = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix}

The determinant of A is calculated as follows:

∣2345∣=2(5)−3(4)=10−12=−2\begin{vmatrix} 2 & 3 \\ 4 & 5 \end{vmatrix} = 2(5) - 3(4) = 10 - 12 = -2

The inverse of A is calculated as follows:

A−1=1−2[2345]A^{-1} = \frac{1}{-2} \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix}

A−1=[52−32−4222]A^{-1} = \begin{bmatrix} \frac{5}{2} & \frac{-3}{2} \\ \frac{-4}{2} & \frac{2}{2} \end{bmatrix}

Discussion

The concept of multiplying a matrix by a scalar is an important one in linear algebra. It is used to scale the matrix and is an essential tool in many applications, including computer graphics, machine learning, and data analysis.

References

  • [1] "Linear Algebra and Its Applications" by Gilbert Strang
  • [2] "Introduction to Linear Algebra" by Gilbert Strang
  • [3] "Linear Algebra" by David C. Lay

Further Reading

For further reading on the topic of linear algebra, we recommend the following resources:

  • [1] "Linear Algebra and Its Applications" by Gilbert Strang
  • [2] "Introduction to Linear Algebra" by Gilbert Strang
  • [3] "Linear Algebra" by David C. Lay

Conclusion

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about multiplying a matrix by a scalar.

Q: What is a scalar?

A: A scalar is a number that is used to multiply a matrix. In this case, the scalar is (1 over the determinant).

Q: What is the determinant of a matrix?

A: The determinant of a matrix is a scalar value that can be calculated from the matrix. It is used to determine the invertibility of the matrix.

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

A: To calculate the determinant of a 2x2 matrix, you can use the following formula:

∣abcd∣=ad−bc\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc

Q: How do I multiply a matrix by a scalar?

A: To multiply a matrix by a scalar, you can multiply each element of the matrix by the scalar.

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.

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

A: To find the inverse of a 2x2 matrix, you can use the following formula:

A−1=1∣abcd∣[abcd]A^{-1} = \frac{1}{\begin{vmatrix} a & b \\ c & d \end{vmatrix}} \begin{bmatrix} a & b \\ c & d \end{bmatrix}

Q: What is the identity matrix?

A: The identity matrix is a matrix that, when multiplied by any other matrix, results in the same matrix.

Q: How do I multiply two matrices together?

A: To multiply two matrices together, you can use the following formula:

[abcd][efgh]=[ae+bgaf+bhce+dgcf+dh]\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} e & f \\ g & h \end{bmatrix} = \begin{bmatrix} ae + bg & af + bh \\ ce + dg & cf + dh \end{bmatrix}

Q: What are some common applications of matrix multiplication?

A: Matrix multiplication has many common applications, including:

  • Computer graphics
  • Machine learning
  • Data analysis
  • Linear algebra

Q: What are some common mistakes to avoid when multiplying matrices?

A: Some common mistakes to avoid when multiplying matrices include:

  • Not checking if the matrices can be multiplied together
  • Not using the correct formula for matrix multiplication
  • Not checking if the result is a valid matrix

Conclusion

In conclusion, multiplying a matrix by a scalar is an important concept in linear algebra. We have answered some of the most frequently asked questions about matrix multiplication and provided some common applications and mistakes to avoid.