Determinants - Problems Practice

In this lecture, we will solve various problems related to determinants.
Determinants are an important topic in mathematics and have wide applications in various fields.
We will go through the step-by-step solutions for different types of determinant problems.
Make sure you understand the basic concepts and formulas before attempting these problems.
Let’s get started with the first problem.

Problem 1: 2x2 Determinant

Find the value of the determinant for the given matrix: $$\begin{bmatrix} 3 & 4 \ 2 & 1 \end{bmatrix}$$

Step 1: Set up the determinant as a 2x2 matrix.
Step 2: Apply the formula for a 2x2 determinant: $ad - bc$.
Step 3: Substitute the values from the matrix: $(3 \times 1) - (4 \times 2)$.
Step 4: Simplify the expression: $3 - 8 = -5$.
Answer: The value of the determinant is -5.

Problem 2: 3x3 Determinant

Find the value of the determinant for the given matrix: $$\begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix}$$

Step 1: Set up the determinant as a 3x3 matrix.
Step 2: Apply the formula for a 3x3 determinant using the cofactor expansion method.
Step 3: Expand along the first row and calculate the sub-determinants.
Step 4: Evaluate the sub-determinants and apply the alternating sign pattern.
Step 5: Multiply the sub-determinants with their respective signs and sum them up.
Answer: The value of the determinant is 0.

Problem 3: Determinant Properties

Simplify the following determinant using properties of determinants: $$\begin{vmatrix} 3 & 4 \ 2 & 7 \end{vmatrix} + 2\begin{vmatrix} 1 & 2 \ 3 & 4 \end{vmatrix}$$

Step 1: Break down the given determinant into individual determinants.
Step 2: Simplify each determinant using the 2x2 determinant formula.
Step 3: Apply the distributive property to add the determinants together.
Answer: The simplified form of the expression is $37$.

Problem 4: Solving Linear Equations with Determinants

Solve the following system of equations using determinants: $$\begin{align*}

3x + 2y &= 5 \

4x - y &= 1 \end{align*}$$

Step 1: Set up the coefficient matrix and the constant matrix.
Step 2: Calculate the determinants of the coefficient matrix and the x-matrix.
Step 3: Apply Cramer’s rule to find the values of x and y.
Answer: The solution is $x = 1$ and $y = 2$.

Problem 5: Finding the Inverse of a Matrix

Find the inverse of the following matrix: $$\begin{bmatrix} 2 & 3 \ 4 & 5 \end{bmatrix}$$

Step 1: Set up the matrix augmented with the identity matrix.
Step 2: Apply row operations to transform the matrix into row-echelon form.
Step 3: Normalize the matrix to get it in reduced row-echelon form.
Step 4: The matrix on the right side is the inverse of the original matrix.
Answer: The inverse of the given matrix is $\begin{bmatrix} -5/2 & 3/2 \ 2 & -1 \end{bmatrix}$.

Problem 6: Determinant of a Product of Matrices

Find the value of the determinant for the product of two matrices: $$\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}\begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix}$$

Step 1: Multiply the two matrices together.
Step 2: Set up the resulting matrix as a determinant.
Step 3: Simplify the determinant using the formula for a 2x2 determinant.
Answer: The value of the determinant is -2.

Problem 7: Determinant of a Transposed Matrix

Find the value of the determinant for the transposed matrix: $$\begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{bmatrix}^T$$

Step 1: Transpose the given matrix.
Step 2: Set up the transposed matrix as a determinant.
Step 3: Apply the formula for a 2x3 determinant to find the value.
Answer: The value of the determinant is $-3$.

Problem 8: Determinants and Linear Independence

Determine whether the following set of vectors is linearly independent: $$\mathbf{v}_1 = \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 4 \ 5 \ 6 \end{bmatrix}, \quad \mathbf{v}_3 = \begin{bmatrix} 2 \ 4 \ 6 \end{bmatrix}$$

Step 1: Set up the matrix with given vectors as columns.
Step 2: Find the determinant of the matrix.
Step 3: If the determinant is non-zero, the vectors are linearly independent. Otherwise, they are linearly dependent.
Answer: The determinant of the matrix is 0, indicating that the vectors are linearly dependent.

Problem 9: Determinant of a Block Matrix

Find the value of the determinant for the following block matrix: $$\begin{bmatrix} \mathbf{A} & \mathbf{B} \ \mathbf{C} & \mathbf{D} \end{bmatrix}$$

Step 1: Calculate the determinant of matrix A and matrix D separately.
Step 2: Find the product of the determinants and apply the alternating sign pattern.
Answer: The value of the determinant is $(-1)^n \cdot \text{det}(\mathbf{A}) \cdot \text{det}(\mathbf{D})$, where n is the order of the matrix.

Problem 10: Cofactor Expansion

Simplify the following determinant using the cofactor expansion method: $$\begin{vmatrix} 1 & 2 & 3 \ 2 & 4 & 6 \ 3 & 6 & 9 \end{vmatrix}$$

Step 1: Expand along the first column and calculate the sub-determinants using the minor method.
Step 2: Apply the alternating sign pattern to the sub-determinants.
Step 3: Multiply the sub-determinants with their respective signs and sum them up.
Answer: The value of the determinant is 0.

Sure! Here are slides 11 to 20 for the lecture on “Determinants - Problems Practice”:

Problem 11: Cramer’s Rule

Use Cramer’s rule to solve the following system of equations: $$\begin{align*}

2x + 3y &= 8 \

4x - y &= 5 \end{align*}$$

Step 1: Set up the coefficient matrix and the constant matrix.
Step 2: Calculate the determinants of the coefficient matrix and the x-matrix.
Step 3: Apply Cramer’s rule to find the values of x and y.
Step 4: Substitute the values of x and y back into the original equations to check the solution.

Problem 12: Adjoint of a Matrix

Find the adjoint of the matrix: $$\begin{bmatrix} 2 & 1 \ 3 & 4 \end{bmatrix}$$

Step 1: Calculate the cofactor matrix of the given matrix.
Step 2: Transpose the cofactor matrix to get the adjoint.
Step 3: The adjoint matrix is the transpose of the cofactor matrix.
Answer: The adjoint of the given matrix is $\begin{bmatrix} 4 & -3 \ -1 & 2 \end{bmatrix}$.

Problem 13: Determinants and Area of a Parallelogram

Find the area of the parallelogram formed by the vectors: $$\mathbf{v}_1 = \begin{bmatrix} 2 \ 3 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} -1 \ 4 \end{bmatrix}$$

Step 1: Set up the matrix with given vectors as columns.
Step 2: Calculate the determinant of the matrix.
Step 3: The absolute value of the determinant gives the area of the parallelogram.
Answer: The area of the parallelogram is 11 square units.

Problem 14: Linear Independence in 3D Space

Determine whether the following set of vectors is linearly independent: $$\mathbf{v}_1 = \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 2 \ 4 \ 6 \end{bmatrix}, \quad \mathbf{v}_3 = \begin{bmatrix} 1 \ 1 \ 1 \end{bmatrix}$$

Step 1: Set up the matrix with given vectors as columns.
Step 2: Find the determinant of the matrix.
Step 3: If the determinant is non-zero, the vectors are linearly independent. Otherwise, they are linearly dependent.
Answer: The determinant of the matrix is 0, indicating that the vectors are linearly dependent.

Problem 15: Determinants and Volume of a Parallelepiped

Find the volume of the parallelepiped formed by the vectors: $$\mathbf{v}_1 = \begin{bmatrix} 2 \ 3 \ 1 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} -1 \ 4 \ 2 \end{bmatrix}, \quad \mathbf{v}_3 = \begin{bmatrix} 3 \ -2 \ 1 \end{bmatrix}$$

Step 1: Set up the matrix with given vectors as columns.
Step 2: Calculate the determinant of the matrix.
Step 3: The absolute value of the determinant gives the volume of the parallelepiped.
Answer: The volume of the parallelepiped is 37 cubic units.

Problem 16: Determinants and Eigenvalues

Find the eigenvalues of the matrix: $$\begin{bmatrix} 4 & 2 \ 1 & 3 \end{bmatrix}$$

Step 1: Set up the characteristic equation of the matrix.
Step 2: Calculate the determinant of the matrix minus lambda times the identity matrix.
Step 3: Solve the characteristic equation to find the eigenvalues.
Answer: The eigenvalues of the given matrix are 1 and 6.

Problem 17: Determinant of a Diagonal Matrix

Find the value of the determinant for the following diagonal matrix: $$\begin{bmatrix} 2 & 0 & 0 \ 0 & -3 & 0 \ 0 & 0 & 5 \end{bmatrix}$$

Step 1: Multiply the diagonal elements of the matrix.
Step 2: The product of the diagonal elements gives the determinant.
Answer: The value of the determinant is -30.

Problem 18: Determinants and Permutations

Find the value of the determinant for the following matrix: $$\begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix}$$

Step 1: Expand along the first row and calculate the sub-determinants.
Step 2: Apply the alternating sign pattern to the sub-determinants.
Step 3: Multiply the sub-determinants with their respective signs and sum them up.
Answer: The value of the determinant is 0.

Problem 19: Determinant of a Upper Triangular Matrix

Find the value of the determinant for the following upper triangular matrix: $$\begin{bmatrix} 4 & 2 & 1 \ 0 & 3 & 2 \ 0 & 0 & 5 \end{bmatrix}$$

Step 1: Multiply the diagonal elements of the matrix.
Step 2: The product of the diagonal elements gives the determinant.
Answer: The value of the determinant is 60.

Problem 20: Determinant of a Singular Matrix

Find the value of the determinant for the following singular matrix: $$\begin{bmatrix} 2 & 3 \ 4 & 6 \end{bmatrix}$$

Step 1: Calculate the determinant of the matrix.
Step 2: If the determinant is 0, the matrix is singular; otherwise, it is non-singular.
Answer: The determinant of the given matrix is 0, indicating that it is a singular matrix.

That’s it for the slides 11 to 20. Continue with the next set of slides to cover more problems on determinants. Sure! Here are slides 21 to 30 for the lecture on “Determinants - Problems Practice”:

Problem 21: Determinant of a Lower Triangular Matrix

Find the value of the determinant for the following lower triangular matrix: $$\begin{bmatrix} 2 & 0 & 0 \ 1 & -3 & 0 \ -2 & 4 & 5 \end{bmatrix}$$

Step 1: Multiply the diagonal elements of the matrix.
Step 2: The product of the diagonal elements gives the determinant.
Answer: The value of the determinant is -30.

Problem 22: Determinant as a Geometric Scaling Factor

Given a matrix transformation: $$\begin{bmatrix} x’ \ y’ \end{bmatrix} = \begin{bmatrix} 2 & 0 \ 0 & 3 \end{bmatrix}\begin{bmatrix} x \ y \end{bmatrix}$$

Step 1: Set up the given transformation matrix.
Step 2: Find the determinant of the matrix.
Step 3: The absolute value of the determinant gives the scaling factor of the transformation.
Answer: The scaling factor of the transformation is 6.

Problem 23: Determinants and Linear Transformations

Given a matrix transformation: $$\begin{bmatrix} x’ \ y’ \end{bmatrix} = \begin{bmatrix} 1 & -2 \ 3 & 2 \end{bmatrix}\begin{bmatrix} x \ y \end{bmatrix}$$

Step 1: Set up the given transformation matrix.
Step 2: Find the determinant of the matrix.
Step 3: If the determinant is non-zero, the transformation is non-singular and preserves linear independence. Otherwise, it collapses or expands space.
Answer: The determinant of the given matrix is 8, indicating that the transformation preserves linear independence.

Problem 24: Determinant and the Area of a Triangle

Find the area of the triangle formed by the points: $$A(3, 5), \quad B(-1, 2), \quad C(6, 4)$$

Step 1: Set up the matrix with the coordinates of the points as rows.
Step 2: Calculate the determinant of the matrix.
Step 3: The absolute value of half the determinant gives the area of the triangle.
Answer: The area of the triangle formed by the given points is 6 square units.

Problem 25: Determinant as a Test for Collinearity

Determine if the points $A(1, 2), B(3, 4),$ and $C(-1, -2)$ are collinear.

Step 1: Set up the matrix with the coordinates of the points as rows.
Step 2: Calculate the determinant of the matrix.
Step 3: If the determinant is 0, the points are collinear; otherwise, they are non-collinear.
Answer: The determinant of the given matrix is 0, indicating that the points are collinear.

Problem 26: Determinants and Homogeneous System of Equations

Determine if the system of equations has a non-trivial solution: $$\begin{align*}

3x + 2y + z &= 0 \

2x - y + z &= 0 \ x - 3y + 4z &= 0 \end{align*}$$

Step 1: Set up the coefficient matrix and the constant matrix.
Step 2: Calculate the determinant of the coefficient matrix.
Step 3: If the determinant is 0, the system has a non-trivial solution; otherwise, it has only a trivial solution.
Answer: The determinant of the coefficient matrix is 0, indicating that the system has a non-trivial solution.

Problem 27: Determinants and Rectangular Matrix

Find the value of the determinant for the following rectangular matrix: $$\begin{bmatrix} 2 & 3 & 4 \ 1 & 2 & 3 \end{bmatrix}$$

Step 1: Expand along the first row and calculate the sub-determinants.
Step 2