Let \[$ A = \left[\begin{array}{lll}1 & 7 & 2\end{array}\right] \$\].Compute The Determinant Of \[$ A \times 4 \times \left[\begin{array}{ccc}0 & 5 & 2 \\ -1 & 4 & 1\end{array}\right] \$\] Using A Cofactor Expansion Along All The Rows

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. In this article, we will explore how to compute the determinant of a matrix using cofactor expansion along all the rows. We will use the given matrix A and a second matrix to demonstrate this process.

Matrix A and the Second Matrix

Let's define the given matrix A as:

$$A = \left[\begin{array}{lll}1 & 7 & 2\end{array}\right]$$

We are also given a second matrix:

$$B = \left[\begin{array}{ccc}0 & 5 & 2 \\ -1 & 4 & 1\end{array}\right]$$

Computing the Determinant of A × 4 × B

To compute the determinant of A × 4 × B, we need to first compute the product of A and 4, and then multiply the result by B.

Step 1: Compute A × 4

To compute A × 4, we can simply multiply each element of A by 4:

$$A \times 4 = \left[\begin{array}{lll}4 & 28 & 8\end{array}\right]$$

Step 2: Compute the Product of A × 4 and B

To compute the product of A × 4 and B, we can use the standard matrix multiplication algorithm:

$$A \times 4 \times B = \left[\begin{array}{lll}4 & 28 & 8\end{array}\right] \times \left[\begin{array}{ccc}0 & 5 & 2 \\ -1 & 4 & 1\end{array}\right]$$

$$A \times 4 \times B = \left[\begin{array}{lll}4(0) + 28(-1) + 8(1) & 4(5) + 28(4) + 8(1) & 4(2) + 28(1) + 8(1)\end{array}\right]$$

$$A \times 4 \times B = \left[\begin{array}{lll}-24 & 148 & 40\end{array}\right]$$

Step 3: Compute the Determinant of A × 4 × B

To compute the determinant of A × 4 × B, we can use the cofactor expansion along any row or column. Let's use the first row:

$$\det(A \times 4 \times B) = (-24) \times \det \left[\begin{array}{cc}148 & 40\end{array}\right] - 148 \times \det \left[\begin{array}{cc}-24 & 40\end{array}\right] + 40 \times \det \left[\begin{array}{cc}-24 & 148\end{array}\right]$$

$$\det(A \times 4 \times B) = (-24) \times (148 \times 40 - 40 \times (-24)) - 148 \times ((-24) \times 40 - 40 \times (-24)) + 40 \times ((-24) \times 148 - 148 \times (-24))$$

$$\det(A \times 4 \times B) = (-24) \times (5920 + 960) - 148 \times (-960 + 960) + 40 \times (-3528 + 3528)$$

$$\det(A \times 4 \times B) = (-24) \times 6880 - 148 \times 0 + 40 \times 0$$

$$\det(A \times 4 \times B) = -165120$$

Conclusion

In this article, we have demonstrated how to compute the determinant of a matrix using cofactor expansion along all the rows. We have used the given matrix A and a second matrix to demonstrate this process. The determinant of A × 4 × B is -165120.

Cofactor Expansion Along All the Rows

To compute the determinant of a matrix using cofactor expansion along all the rows, we can follow these steps:

1. Choose a row to expand along.
2. Compute the determinant of the submatrix obtained by removing the row and column of the chosen element.
3. Multiply the chosen element by the determinant of the submatrix and add or subtract it from the sum of the products of the other elements and their cofactors.
4. Repeat steps 2-3 for each element in the chosen row.
5. The final result is the determinant of the matrix.

Example

Let's use the matrix A and the second matrix B to demonstrate the cofactor expansion along all the rows:

$$A = \left[\begin{array}{lll}1 & 7 & 2\end{array}\right]$$

$$B = \left[\begin{array}{ccc}0 & 5 & 2 \\ -1 & 4 & 1\end{array}\right]$$

To compute the determinant of A × 4 × B, we can use the cofactor expansion along the first row:

$$\det(A \times 4 \times B) = (-24) \times \det \left[\begin{array}{cc}148 & 40\end{array}\right] - 148 \times \det \left[\begin{array}{cc}-24 & 40\end{array}\right] + 40 \times \det \left[\begin{array}{cc}-24 & 148\end{array}\right]$$

$$\det(A \times 4 \times B) = (-24) \times (148 \times 40 - 40 \times (-24)) - 148 \times ((-24) \times 40 - 40 \times (-24)) + 40 \times ((-24) \times 148 - 148 \times (-24))$$

$$\det(A \times 4 \times B) = (-24) \times (5920 + 960) - 148 \times (-960 + 960) + 40 \times (-3528 + 3528)$$

$$\det(A \times 4 \times B) = (-24) \times 6880 - 148 \times 0 + 40 \times 0$$

$$\det(A \times 4 \times B) = -165120$$

Conclusion

In this article, we have demonstrated how to compute the determinant of a matrix using cofactor expansion along all the rows. We have used the given matrix A and a second matrix to demonstrate this process. The determinant of A × 4 × B is -165120.

References

• [1] Linear Algebra and Its Applications, 4th Edition, by Gilbert Strang
• [2] Matrix Algebra, 2nd Edition, by James E. Gentle
• [3] Determinants and Matrices, by Charles W. Curtis
  Determinant Computation Using Cofactor Expansion: Q&A =====================================================

Introduction

In our previous article, we explored how to compute the determinant of a matrix using cofactor expansion along all the rows. In this article, we will answer some frequently asked questions about determinant computation using cofactor expansion.

Q: What is cofactor expansion?

A: Cofactor expansion is a method of computing the determinant of a matrix by expanding along a row or column. It involves multiplying each element in the chosen row or column by its cofactor and summing the results.

Q: What is a cofactor?

A: A cofactor is the determinant of the submatrix obtained by removing the row and column of the chosen element. It is denoted by Cij, where i is the row number and j is the column number.

Q: How do I choose the row or column to expand along?

A: You can choose any row or column to expand along. However, it is often easier to choose a row or column with the most zeros, as this will simplify the computation.

Q: What is the formula for cofactor expansion?

A: The formula for cofactor expansion along a row is:

$$\det(A) = a_{11}C_{11} + a_{12}C_{12} + ... + a_{1n}C_{1n}$$

where aij is the element in the ith row and jth column, and Cij is the cofactor of aij.

Q: How do I compute the cofactor of an element?

A: To compute the cofactor of an element, you need to remove the row and column of the element and compute the determinant of the resulting submatrix.

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

A: The determinant of a matrix is equal to the sum of the products of the elements and their cofactors.

Q: Can I use cofactor expansion to compute the determinant of a matrix with complex numbers?

A: Yes, you can use cofactor expansion to compute the determinant of a matrix with complex numbers. However, you need to be careful when computing the cofactors, as the determinant of a matrix with complex numbers can be complex.

Q: Is cofactor expansion the only method for computing the determinant of a matrix?

A: No, there are other methods for computing the determinant of a matrix, such as the Laplace expansion and the LU decomposition. However, cofactor expansion is a simple and efficient method that can be used for small to medium-sized matrices.

Q: What are some common applications of determinant computation using cofactor expansion?

A: Determinant computation using cofactor expansion has many applications in linear algebra, including:

• Computing the inverse of a matrix
• Solving systems of linear equations
• Computing the eigenvalues and eigenvectors of a matrix
• Computing the rank of a matrix

Conclusion

In this article, we have answered some frequently asked questions about determinant computation using cofactor expansion. We hope that this article has provided you with a better understanding of this important topic in linear algebra.

References

• [1] Linear Algebra and Its Applications, 4th Edition, by Gilbert Strang
• [2] Matrix Algebra, 2nd Edition, by James E. Gentle
• [3] Determinants and Matrices, by Charles W. Curtis

Additional Resources

• [1] Khan Academy: Determinants and Cofactors
• [2] MIT OpenCourseWare: Linear Algebra
• [3] Wolfram MathWorld: Determinant

Practice Problems

1. Compute the determinant of the matrix A using cofactor expansion along the first row.

$$A = \left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right]$$

2. Compute the determinant of the matrix B using cofactor expansion along the second column.

$$B = \left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right]$$

3. Compute the determinant of the matrix C using cofactor expansion along the third row.

$$C = \left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right]$$

Answers

1. $\det(A) = 1 \times C_{11} + 2 \times C_{12} + 3 \times C_{13} = 1 \times (5 \times 9 - 6 \times 8) + 2 \times (4 \times 9 - 6 \times 7) + 3 \times (4 \times 8 - 5 \times 7) = 1 \times 27 + 2 \times 18 + 3 \times 11 = 27 + 36 + 33 = 96$

2. $\det(B) = 2 \times C_{21} + 5 \times C_{22} + 8 \times C_{23} = 2 \times (1 \times 9 - 3 \times 7) + 5 \times (1 \times 6 - 3 \times 4) + 8 \times (1 \times 5 - 2 \times 4) = 2 \times (-6) + 5 \times (-3) + 8 \times (-3) = -12 - 15 - 24 = -51$

3. $\det(C) = 7 \times C_{31} + 8 \times C_{32} + 9 \times C_{33} = 7 \times (1 \times 5 - 2 \times 4) + 8 \times (1 \times 6 - 2 \times 3) + 9 \times (1 \times 8 - 2 \times 5) = 7 \times (-3) + 8 \times (0) + 9 \times (-2) = -21 - 18 = -39$