Matrix and Determinant - Problems on determinants and inverse of matrix

Slide 1

  • In this lecture, we will solve some problems related to determinants and inverse of a matrix.

Slide 2

  • Problem 1: Find the determinant of the matrix A = [[3, 2], [4, 5]].
    • Solution: The determinant of matrix A is given by the formula: |A| = ad - bc
    • Substitute the values from matrix A: |A| = (3 * 5) - (2 * 4) = 15 - 8 = 7

Slide 3

  • Problem 2: Solve the system of equations using Cramer’s rule:
    • 2x + 3y = 7
    • 4x - 5y = 11
    • Solution: First, find the determinant of the coefficient matrix D. Then, find the determinant of Dx and Dy, where Dx and Dy are obtained by replacing the columns of D with the respective right-hand side column matrix.

Slide 4

  • Problem 3: Find the inverse of matrix B = [[1, 2], [3, 4]].
    • Solution: The inverse of a matrix B is given by the formula: B^(-1) = (1/|B|) * adj(B)
    • Find the determinant |B| = (1 * 4) - (2 * 3) = 4 - 6 = -2
    • Find the adjoint matrix adj(B) = [[4, -2], [-3, 1]]
    • Calculate the inverse B^(-1) = (1/(-2)) * [[4, -2], [-3, 1]] = [[-2, 1], [3/2, -1/2]]

Slide 5

  • Problem 4: Solve the following system of equations using matrix inversion method:
    • 3x + 2y = 5
    • 4x - 3y = 7
    • Solution: Write the system of equations in matrix form as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the column matrix of constants. Find the inverse of matrix A and multiply it with matrix B to obtain the solution matrix X.

Slide 6

  • Problem 5: Find the value of k for which the matrix C = [[1, 2], [4, k]] is non-invertible (singular).
    • Solution: The determinant of matrix C must be zero for it to be singular. Calculate |C| = (1 * k) - (2 * 4) = k - 8. Set k - 8 = 0, solve for k to find the value for which the matrix is singular.

Slide 7

  • Problem 6: Find the value of x for which the matrix D = [[2, 1], [x, 3]] is non-invertible (singular).
    • Solution: The determinant of matrix D must be zero for it to be singular. Calculate |D| = (2 * 3) - (1 * x) = 6 - x. Set 6 - x = 0, solve for x to find the value for which the matrix is singular.

Slide 8

  • Problem 7: Solve the system of equations using matrix inversion method:
    • 2x + y + z = 9
    • x + 3y - z = -4
    • 3x - y + 2z = 18
    • Solution: Write the system of equations in matrix form as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the column matrix of constants. Find the inverse of matrix A and multiply it with matrix B to obtain the solution matrix X.

Slide 9

  • Problem 8: Find the inverse of matrix E = [[2, 1, 4], [3, 0, 1], [5, 2, 3]].
    • Solution: The inverse of a matrix E is given by the formula: E^(-1) = (1/|E|) * adj(E)
    • Find the determinant |E| by using cofactor expansion or row reduction methods.
    • Find the adjoint matrix adj(E) by calculating the cofactor matrix of E and taking its transpose.
    • Calculate the inverse E^(-1) = (1/|E|) * adj(E)

Slide 10

  • Problem 9: Solve the following system of equations using matrix inversion:
    • 2x + 3y + z = 9
    • x + 2y - z = 1
    • 3x + y + 2z = 5
    • Solution: Write the system of equations in matrix form as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the column matrix of constants. Find the inverse of matrix A and multiply it with matrix B to obtain the solution matrix X.

Slide 11

  • Problem 10: Find the inverse of matrix F = [[1, -2, 3], [-1, 3, -4], [2, -5, 7]].
    • Solution: The inverse of a matrix F is given by the formula: F^(-1) = (1/|F|) * adj(F)
    • Find the determinant |F| by using cofactor expansion or row reduction methods.
    • Find the adjoint matrix adj(F) by calculating the cofactor matrix of F and taking its transpose.
    • Calculate the inverse F^(-1) = (1/|F|) * adj(F)

Slide 12

  • Problem 11: Solve the following system of equations using matrix inversion:
    • 3x - y = 5
    • 4x + 2y = 10
    • Solution: Write the system of equations in matrix form as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the column matrix of constants. Find the inverse of matrix A and multiply it with matrix B to obtain the solution matrix X.

Slide 13

  • Problem 12: Find the determinant of the matrix G = [[2, 3, 1], [5, -2, 4], [0, 1, 3]] using the expansion method.
    • Solution: The determinant of a 3x3 matrix G can be calculated using the expansion method.
    • Expand along the first row or column and calculate the determinants of the 2x2 submatrices formed.
    • Use the pattern (+ - +) while expanding to simplify the calculation.

Slide 14

  • Problem 13: Solve the following system of equations using matrix inversion:
    • 2x + 4y - z = 7
    • 3x - 2y + 2z = 10
    • x + 3y - 2z = 0
    • Solution: Write the system of equations in matrix form as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the column matrix of constants. Find the inverse of matrix A and multiply it with matrix B to obtain the solution matrix X.

Slide 15

  • Problem 14: Find the inverse of matrix H = [[3, 0, 2], [-1, 2, 1], [4, 2, 5]].
    • Solution: The inverse of a matrix H is given by the formula: H^(-1) = (1/|H|) * adj(H)
    • Find the determinant |H| by using cofactor expansion or row reduction methods.
    • Find the adjoint matrix adj(H) by calculating the cofactor matrix of H and taking its transpose.
    • Calculate the inverse H^(-1) = (1/|H|) * adj(H)

Slide 16

  • Problem 15: Solve the following system of equations using matrix inversion:
    • 5x + 3y - z = 2
    • 2x + 2y + 4z = 6
    • 3x - 2y + 3z = 1
    • Solution: Write the system of equations in matrix form as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the column matrix of constants. Find the inverse of matrix A and multiply it with matrix B to obtain the solution matrix X.

Slide 17

  • Problem 16: Find the inverse of matrix I = [[1, -1, 2], [0, 3, -1], [2, 1, 4]].
    • Solution: The inverse of a matrix I is given by the formula: I^(-1) = (1/|I|) * adj(I)
    • Find the determinant |I| by using cofactor expansion or row reduction methods.
    • Find the adjoint matrix adj(I) by calculating the cofactor matrix of I and taking its transpose.
    • Calculate the inverse I^(-1) = (1/|I|) * adj(I)

Slide 18

  • Problem 17: Solve the following system of equations using matrix inversion:
    • 2x + y + 3z = 6
    • x - 2y + z = 3
    • 3x + y - 2z = 0
    • Solution: Write the system of equations in matrix form as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the column matrix of constants. Find the inverse of matrix A and multiply it with matrix B to obtain the solution matrix X.

Slide 19

  • Problem 18: Find the determinant of the matrix J = [[1, 2, 3, 4], [0, 1, 2, 3], [1, 1, 1, 1], [2, 2, 2, 2]] using the expansion method.
    • Solution: The determinant of a 4x4 matrix J can be calculated using the expansion method.
    • Expand along the first row or column and calculate the determinants of the 3x3 submatrices formed.
    • Use the pattern (+ - + -) while expanding to simplify the calculation.

Slide 20

  • Problem 19: Solve the following system of equations using matrix inversion:
    • x + 2y - z + w = 6
    • 2x + y + w = 4
    • 3x - y + 2z - w = 8
    • x + y + z + w = 5
    • Solution: Write the system of equations in matrix form as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the column matrix of constants. Find the inverse of matrix A and multiply it with matrix B to obtain the solution matrix X.

Slide 21

  • Problem 20: Find the inverse of matrix K = [[1, -1, 2, 0], [0, 1, -1, 3], [2, 3, 4, -1], [1, 2, 3, 0]].
    • Solution: The inverse of a matrix K is given by the formula: K^(-1) = (1/|K|) * adj(K)
    • Find the determinant |K| by using cofactor expansion or row reduction methods.
    • Find the adjoint matrix adj(K) by calculating the cofactor matrix of K and taking its transpose.
    • Calculate the inverse K^(-1) = (1/|K|) * adj(K)

Slide 22

  • Problem 21: Solve the following system of equations using matrix inversion:
    • x + 2y + 3z - w = 5
    • 2x + y - 2z + 4w = 9
    • 3x - y + z + 2w = 1
    • x + 3y - 4z + 2w = 7
    • Solution: Write the system of equations in matrix form as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the column matrix of constants. Find the inverse of matrix A and multiply it with matrix B to obtain the solution matrix X.

Slide 23

  • Problem 22: Find the determinant of the matrix L = [[1, 1, 0, 1], [2, -1, 3, 2], [1, 2, 1, 0], [1, -1, 4, 5]] using the expansion method.
    • Solution: The determinant of a 4x4 matrix L can be calculated using the expansion method.
    • Expand along the first row or column and calculate the determinants of the 3x3 submatrices formed.
    • Use the pattern (+ - + -) while expanding to simplify the calculation.

Slide 24

  • Problem 23: Solve the following system of equations using matrix inversion:
    • 2x + y + z = 3
    • -x + 3y - z = 4
    • 4x - 2y + z = -1
    • Solution: Write the system of equations in matrix form as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the column matrix of constants. Find the inverse of matrix A and multiply it with matrix B to obtain the solution matrix X.

Slide 25

  • Problem 24: Find the inverse of matrix M = [[1, -1, 0, 2, -1], [0, 3, 1, -1, 0], [2, 0, -2, 1, 3], [1, -1, 0, 2, 1], [0, 2, 1, -1, 1]].
    • Solution: The inverse of a matrix M is given by the formula: M^(-1) = (1/|M|) * adj(M)
    • Find the determinant |M| by using cofactor expansion or row reduction methods.
    • Find the adjoint matrix adj(M) by calculating the cofactor matrix of M and taking its transpose.
    • Calculate the inverse M^(-1) = (1/|M|) * adj(M)

Slide 26

  • Problem 25: Solve the following system of equations using matrix inversion:
    • x + 2y - z = -3
    • 2x - y + 3z = 7
    • 3x + y + 2z = 12
    • Solution: Write the system of equations in matrix form as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the column matrix of constants. Find the inverse of matrix A and multiply it with matrix B to obtain the solution matrix X.

Slide 27

  • Problem 26: Find the determinant of the matrix N = [[1, 2, 3, 4, 5], [0, 1, 2, 3, 4], [1, 0, 1, 0, 1], [2, 2, 2, 2, 2], [1, 1, 0, 1, 0]] using the expansion method.
    • Solution: The determinant of a 5x5 matrix N can be calculated using the expansion method.
    • Expand along the first row or column and calculate the determinants of the 4x4 submatrices formed.
    • Use the pattern (+ - + - +) while expanding to simplify the calculation.

Slide 28

  • Problem 27: Solve the following system of equations using matrix inversion:
    • x + y + 2z - w = 5
    • -x + y - z + w = 3
    • 3x + 2y + 3z + 2w = 13
    • x + y - z + 3w = 9
    • Solution: Write the system of equations in matrix form as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the column matrix of constants. Find the inverse of matrix A and multiply it with matrix B to obtain the solution matrix X.

Slide 29

  • Problem 28: Find the inverse of matrix O = [[1, 0, 2, 1, -1], [0, 2, -1, 1, 0], [3, 2, 1, 0, 1], [-1, 3, 1, -2, 0], [1, 0, 3, 1, -1]].
    • Solution: The inverse of a matrix O is given by the formula: O^(-1) = (1/|O|) * adj(O)
    • Find the determinant |O| by using cofactor expansion or row reduction methods.
    • Find the adjoint matrix adj(O) by calculating the cofactor matrix of O and taking its transpose.
    • Calculate the inverse O^(-1) = (1/|O|) * adj(O)

Slide 30

  • Problem 29: Solve the following system of equations using matrix inversion:
    • x + 3y - z + 2w = 6
    • 2x - y + 4z + w = 5
    • 3x + y + 2z - w = 1
    • x - 2y + 3z + 4w = 3
    • Solution: Write the system of equations in matrix form as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the column matrix of constants. Find the inverse of matrix A and multiply it with matrix B to obtain the solution matrix X.