2. Evaluate The Determinant Of The Matrix By First Reducing The Matrix To Row Echelon Form And Then Using Some Combination Of Row Operations And Cofactor Expansion.(i) $\[ \left\lvert\, \begin{array}{rrr} 3 & -6 & 9 \\ -2 & 7 & -2 \\ 0 & 1 &

Introduction

Determinants are a fundamental concept in linear algebra, and understanding how to evaluate them is crucial for solving systems of linear equations and other mathematical problems. In this article, we will explore two methods for evaluating the determinant of a matrix: reducing the matrix to row echelon form and using a combination of row operations and cofactor expansion.

Method 1: Reducing the Matrix to Row Echelon Form

One way to evaluate the determinant of a matrix is to reduce it to row echelon form (REF) using a series of row operations. The row echelon form of a matrix is a simplified form where all the entries below the leading entry of each row are zero.

Step 1: Perform Row Operations to Obtain Row Echelon Form

To reduce the matrix to row echelon form, we need to perform a series of row operations. These operations include:

• Swap two rows: Swap rows i and j.
• Multiply a row by a scalar: Multiply row i by a scalar k.
• Add a multiple of one row to another: Add k times row i to row j.

We will use these operations to transform the given matrix into row echelon form.

Step 2: Evaluate the Determinant

Once we have reduced the matrix to row echelon form, we can evaluate the determinant using the following formula:

• Determinant of a triangular matrix: If the matrix is triangular (either upper or lower), the determinant is the product of the entries on the main diagonal.

Example

Let's consider the following matrix:

$$\left\lvert\, \begin{array}{rrr} 3 & -6 & 9 \\ -2 & 7 & -2 \\ 0 & 1 & 4 \end{array} \right\lvert$$

To reduce this matrix to row echelon form, we can perform the following row operations:

• Swap rows 1 and 2.
• Multiply row 2 by -1/3.
• Add 2 times row 2 to row 3.

Performing these operations, we obtain the following matrix in row echelon form:

$$\left\lvert\, \begin{array}{rrr} -2 & 7 & -2 \\ 3 & -6 & 9 \\ 0 & 1 & 4 \end{array} \right\lvert$$

Now, we can evaluate the determinant using the formula for the determinant of a triangular matrix:

$$\det(A) = (-2) \times (-6) \times 4 = 48$$

Method 2: Using a Combination of Row Operations and Cofactor Expansion

Another way to evaluate the determinant of a matrix is to use a combination of row operations and cofactor expansion.

Step 1: Perform Row Operations to Simplify the Matrix

We can perform row operations to simplify the matrix and make it easier to evaluate the determinant.

Step 2: Evaluate the Determinant Using Cofactor Expansion

Once we have simplified the matrix, we can evaluate the determinant using cofactor expansion. The cofactor expansion formula is:

• Cofactor expansion: The determinant of a matrix A can be evaluated using the cofactor expansion formula:

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

where $C_{ij}$ is the cofactor of the entry $a_{ij}$.

Example

Let's consider the following matrix:

$$\left\lvert\, \begin{array}{rrr} 3 & -6 & 9 \\ -2 & 7 & -2 \\ 0 & 1 & 4 \end{array} \right\lvert$$

To evaluate the determinant using cofactor expansion, we can perform the following row operations:

• Swap rows 1 and 2.
• Multiply row 2 by -1/3.
• Add 2 times row 2 to row 3.

Performing these operations, we obtain the following matrix:

$$\left\lvert\, \begin{array}{rrr} -2 & 7 & -2 \\ 3 & -6 & 9 \\ 0 & 1 & 4 \end{array} \right\lvert$$

Now, we can evaluate the determinant using cofactor expansion:

$$\det(A) = (-2) \times C_{11} + 3 \times C_{12} + 0 \times C_{13}$$

where $C_{ij}$ is the cofactor of the entry $a_{ij}$.

To evaluate the cofactors, we need to find the determinant of the submatrices obtained by removing the row and column of the entry $a_{ij}$.

Evaluating Cofactors

To evaluate the cofactors, we need to find the determinant of the submatrices obtained by removing the row and column of the entry $a_{ij}$.

For example, to evaluate the cofactor $C_{11}$, we need to find the determinant of the submatrix obtained by removing the row and column of the entry $a_{11}$:

$$\left\lvert\, \begin{array}{rr} 7 & -2 \\ 1 & 4 \end{array} \right\lvert$$

The determinant of this submatrix is:

$$\det(A_{11}) = 7 \times 4 - (-2) \times 1 = 30$$

Now, we can evaluate the cofactor $C_{11}$:

$$C_{11} = (-1)^{1+1} \times \det(A_{11}) = 30$$

Similarly, we can evaluate the cofactors $C_{12}$ and $C_{13}$:

$$C_{12} = (-1)^{1+2} \times \det(A_{12}) = -(-2) \times 4 = 8$$

$$C_{13} = (-1)^{1+3} \times \det(A_{13}) = 3 \times 1 = 3$$

Now, we can evaluate the determinant using cofactor expansion:

$$\det(A) = (-2) \times 30 + 3 \times 8 + 0 \times 3 = 48$$

Conclusion

In this article, we have explored two methods for evaluating the determinant of a matrix: reducing the matrix to row echelon form and using a combination of row operations and cofactor expansion. We have also provided examples to illustrate these methods.

The determinant of a matrix is a fundamental concept in linear algebra, and understanding how to evaluate it is crucial for solving systems of linear equations and other mathematical problems.

References

• Linear Algebra and Its Applications by Gilbert Strang
• Introduction to Linear Algebra by Jim Hefferon
• Determinants and Matrices by Michael Artin

Further Reading

• Linear Algebra and Its Applications by Gilbert Strang
• Introduction to Linear Algebra by Jim Hefferon
• Determinants and Matrices by Michael Artin

Glossary

• Determinant: A scalar value that can be computed from the entries of a square matrix.
• Row echelon form: A simplified form of a matrix where all the entries below the leading entry of each row are zero.
• Cofactor expansion: A method for evaluating the determinant of a matrix using the cofactors of the entries.
• Cofactor: The determinant of the submatrix obtained by removing the row and column of the entry.
  Determinants: A Q&A Guide ==========================

Introduction

Determinants are a fundamental concept in linear algebra, and understanding how to evaluate them is crucial for solving systems of linear equations and other mathematical problems. In this article, we will answer some frequently asked questions about determinants.

Q: What is a determinant?

A: A determinant is a scalar value that can be computed from the entries of a square matrix. It is a fundamental concept in linear algebra and is used to solve systems of linear equations.

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

A: There are several methods for evaluating the determinant of a matrix, including:

• Reducing the matrix to row echelon form: This involves performing a series of row operations to simplify the matrix and make it easier to evaluate the determinant.
• Using a combination of row operations and cofactor expansion: This involves performing a series of row operations to simplify the matrix and then using cofactor expansion to evaluate the determinant.

Q: What is row echelon form?

A: Row echelon form is a simplified form of a matrix where all the entries below the leading entry of each row are zero. This form is useful for evaluating the determinant of a matrix.

Q: What is cofactor expansion?

A: Cofactor expansion is a method for evaluating the determinant of a matrix using the cofactors of the entries. The cofactor of an entry is the determinant of the submatrix obtained by removing the row and column of the entry.

Q: How do I evaluate the cofactors of a matrix?

A: To evaluate the cofactors of a matrix, you need to find the determinant of the submatrices obtained by removing the row and column of the entry. This can be done using the formula for the determinant of a 2x2 matrix.

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

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

$$\det(A) = a_{11}a_{22} - a_{12}a_{21}$$

Q: How do I use cofactor expansion to evaluate the determinant of a matrix?

A: To use cofactor expansion to evaluate the determinant of a matrix, you need to:

• Perform a series of row operations to simplify the matrix: This will make it easier to evaluate the determinant.
• Evaluate the cofactors of the entries: This will involve finding the determinant of the submatrices obtained by removing the row and column of the entry.
• Use the cofactor expansion formula to evaluate the determinant: This will involve using the cofactors of the entries to evaluate the determinant.

Q: What are some common mistakes to avoid when evaluating the determinant of a matrix?

A: Some common mistakes to avoid when evaluating the determinant of a matrix include:

• Not simplifying the matrix enough: This can make it difficult to evaluate the determinant.
• Not evaluating the cofactors correctly: This can lead to incorrect results.
• Not using the correct formula for the determinant: This can lead to incorrect results.

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

A: A matrix is invertible if and only if its determinant is non-zero. This means that if the determinant of a matrix is zero, the matrix is not invertible.

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

A: The determinant of a matrix and its inverse are related by the following formula:

$$\det(A^{-1}) = \frac{1}{\det(A)}$$

Conclusion

In this article, we have answered some frequently asked questions about determinants. We have covered topics such as how to evaluate the determinant of a matrix, what is row echelon form, what is cofactor expansion, and how to use cofactor expansion to evaluate the determinant of a matrix. We have also covered some common mistakes to avoid when evaluating the determinant of a matrix and the relationship between the determinant of a matrix and its inverse.

References

• Linear Algebra and Its Applications by Gilbert Strang
• Introduction to Linear Algebra by Jim Hefferon
• Determinants and Matrices by Michael Artin

Further Reading

• Linear Algebra and Its Applications by Gilbert Strang
• Introduction to Linear Algebra by Jim Hefferon
• Determinants and Matrices by Michael Artin

Glossary

• Determinant: A scalar value that can be computed from the entries of a square matrix.
• Row echelon form: A simplified form of a matrix where all the entries below the leading entry of each row are zero.
• Cofactor expansion: A method for evaluating the determinant of a matrix using the cofactors of the entries.
• Cofactor: The determinant of the submatrix obtained by removing the row and column of the entry.