Matrix and Determinant

• Properties of Determinants
• The determinant of a matrix can be used to solve various problems in linear algebra.
• Today, we will focus on solving problems related to the properties of determinants.
• Let’s dive into some examples!

Example 1

Given a matrix A = [[a, b], [c, d]], find the value of |A|.

• The determinant of a 2x2 matrix A = [[a, b], [c, d]] is given by |A| = ad - bc.
• Substitute the values from the matrix: |A| = ad - bc.
• Solve the equation and find the value of |A|.
• Let’s solve a numerical example for better understanding.

Example 2

Given a matrix B = [[3,-1,2], [0,5,4], [1,2,0]], find the value of |B|.

• The determinant of a 3x3 matrix B = [[3,-1,2], [0,5,4], [1,2,0]] can be calculated using the cofactor expansion method.
• We need to express the given matrix as a sum of products of the elements and their respective cofactors.
• Let’s substitute the values and solve the equation.
• Calculate the determinants of the 2x2 matrices formed from the elements of B.
• Use the formula |B| = 3a - b + 2c to find the final value of |B|.

Example 3

Given a matrix C = [[2,0,1,3], [1,5,2,0], [0,3,4,1], [2,1,0,2]], find the value of |C|.

• The determinant of a 4x4 matrix C can also be calculated using the cofactor expansion method.
• Express the given matrix as a sum of products of the elements and their respective cofactors.
• Substitute the values and solve the equation.
• Calculate the determinants of the 3x3 matrices formed from the elements of C.
• Use the formula |C| = 2a - 0b + c - 3d to find the final value of |C|.

Properties of Determinants

1. The determinant of a matrix remains unchanged when its rows and columns are interchanged.

2. If two rows or columns of a matrix are identical, the determinant of that matrix is zero.

3. If we multiply each element of a row or column of a matrix by a scalar ‘k’, the determinant of the resulting matrix is ‘k’ times the determinant of the original matrix.

4. If we add a scalar multiple of one row or column to another row or column, the determinant of the resulting matrix remains unchanged.

5. If a matrix has a row or column consisting of zeros, its determinant is zero.

6. The determinant of a triangular matrix is the product of its diagonal elements.

Example 4

Given a matrix D = [[2,1,3], [-1,0,2], [4,2,1]], find the value of |D|.

• Apply the properties of determinants to simplify the calculation.
• Interchange the rows or columns if it helps simplify the calculation.
• Use the formula |D| = a(ei - fh) - b(di - fg) + c(dh - eg) to find the final value of |D|.

Example 5

Given a matrix E = [[1,2,0], [0,1,3], [2,3,1]], find the value of |E|.

• Apply the properties of determinants to simplify the calculation.
• Use the formula |E| = a(ei - fh) - b(di - fg) + c(dh - eg) to find the final value of |E|.
• Substituting the values, calculate the determinants of the 2x2 matrices formed from the elements of E.
• Solve the equation and find the value of |E|.

Matrix and Determinant - Problem on properties of determinants

Problem 1

Given a matrix F = [[1,3], [2,4]], find the value of |F|.

• Apply the formula |F| = ad - bc to find the determinant of F.
• Substitute the values and solve the equation.
• Calculate the determinants of the 2x2 matrices formed from the elements of F.
• Use the formula |F| = 1(4) - 3(2) to find the final value of |F|.

Problem 2

Given a matrix G = [[-3,4,2], [1,2,-1], [5,0,3]], find the value of |G|.

• Apply the properties of determinants to simplify the calculation.
• Use the formula |G| = a(ei - fh) - b(di - fg) + c(dh - eg) to find the final value of |G|.
• Calculate the determinants of the 2x2 matrices formed from the elements of G.
• Solve the equation and find the value of |G|.

Problem 3

Given a matrix H = [[-1,3,-2,4], [2,0,-3,1], [0,1,2,-2], [3,-1,0,2]], find the value of |H|.

• Apply the properties of determinants to simplify the calculation.
• Use the formula |H| = a(ei - fh) - b(di - fg) + c(dh - eg) to find the final value of |H|.
• Calculate the determinants of the 3x3 matrices formed from the elements of H.
• Solve the equation and find the value of |H|.

Problem 4

Given a matrix J = [[1,0,3], [2,-1,4], [-2,3,-1]], find the value of |J|.

• Apply the properties of determinants to simplify the calculation.
• Use the formula |J| = a(ei - fh) - b(di - fg) + c(dh - eg) to find the final value of |J|.
• Calculate the determinants of the 2x2 matrices formed from the elements of J.
• Solve the equation and find the value of |J|.

Problem 5

Given a matrix K = [[1,2,3,4], [5,6,7,8], [9,10,11,12], [13,14,15,16]], find the value of |K|.

• Apply the properties of determinants to simplify the calculation.
• Use the formula |K| = a(ei - fh) - b(di - fg) + c(dh - eg) to find the final value of |K|.
• Calculate the determinants of the 3x3 matrices formed from the elements of K.
• Solve the equation and find the value of |K|.

Problem 6

Given a matrix L = [[-2,1,3,4], [2,-1,-3,0], [0,-2,1,5], [1,0,4,-2]], find the value of |L|.

• Apply the properties of determinants to simplify the calculation.
• Use the formula |L| = a(ei - fh) - b(di - fg) + c(dh - eg) to find the final value of |L|.
• Calculate the determinants of the 3x3 matrices formed from the elements of L.
• Solve the equation and find the value of |L|.

Problem 7

Given a matrix M = [[1,2,3,4], [0,1,0,1], [2,1,-1,2], [3,4,-2,1]], find the value of |M|.

• Apply the properties of determinants to simplify the calculation.
• Use the formula |M| = a(ei - fh) - b(di - fg) + c(dh - eg) to find the final value of |M|.
• Calculate the determinants of the 3x3 matrices formed from the elements of M.
• Solve the equation and find the value of |M|.

Problem 8

Given a matrix N = [[1,2,3,4], [-1,0,1,2], [0,0,-1,-2], [2,1,1,2]], find the value of |N|.

• Apply the properties of determinants to simplify the calculation.
• Use the formula |N| = a(ei - fh) - b(di - fg) + c(dh - eg) to find the final value of |N|.
• Calculate the determinants of the 3x3 matrices formed from the elements of N.
• Solve the equation and find the value of |N|.

Problem 9

Given a matrix P = [[1,2,3,4,5], [0,2,0,1,2], [-1,0,1,0,1], [3,4,2,1,2], [1,2,3,4,5]], find the value of |P|.

• Apply the properties of determinants to simplify the calculation.
• Use the formula |P| = a(ei - fh) - b(di - fg) + c(dh - eg) to find the final value of |P|.
• Calculate the determinants of the 4x4 matrices formed from the elements of P.
• Solve the equation and find the value of |P|.

21. Matrix and Determinant - Problem on properties of determinants

• Problem 10
  • Given a matrix Q = [[3,1,-2,0], [0,2,-1,3], [1,0,2,1], [2,-1,3,4]], find the value of |Q|.
  • Apply the properties of determinants to simplify the calculation.
  • Use the formula |Q| = a(ei - fh) - b(di - fg) + c(dh - eg) to find the final value of |Q|.
  • Calculate the determinants of the 3x3 matrices formed from the elements of Q.
  • Solve the equation and find the value of |Q|.

22. Matrix and Determinant - Problem on properties of determinants

• Problem 11
  • Given a matrix R = [[-2,-1,3,5], [1,0,-2,-4], [-1,2,3,1], [3,5,7,2]], find the value of |R|.
  • Apply the properties of determinants to simplify the calculation.
  • Use the formula |R| = a(ei - fh) - b(di - fg) + c(dh - eg) to find the final value of |R|.
  • Calculate the determinants of the 3x3 matrices formed from the elements of R.
  • Solve the equation and find the value of |R|.

23. Matrix and Determinant - Problem on properties of determinants

• Problem 12
  • Given a matrix S = [[1,-1,2,3], [2,2,3,5], [-1,3,1,4], [3,0,2,1]], find the value of |S|.
  • Apply the properties of determinants to simplify the calculation.
  • Use the formula |S| = a(ei - fh) - b(di - fg) + c(dh - eg) to find the final value of |S|.
  • Calculate the determinants of the 3x3 matrices formed from the elements of S.
  • Solve the equation and find the value of |S|.

24. Matrix and Determinant - Problem on properties of determinants

• Problem 13
  • Given a matrix T = [[1,2,3], [2,3,1], [3,1,2]], find the value of |T|.
  • Apply the properties of determinants to simplify the calculation.
  • Use the formula |T| = a(ei - fh) - b(di - fg) + c(dh - eg) to find the final value of |T|.
  • Calculate the determinants of the 2x2 matrices formed from the elements of T.
  • Solve the equation and find the value of |T|.

25. Matrix and Determinant - Problem on properties of determinants

• Problem 14
  • Given a matrix U = [[4,1,-2], [-1,3,2], [2,0,5]], find the value of |U|.
  • Apply the properties of determinants to simplify the calculation.
  • Use the formula |U| = a(ei - fh) - b(di - fg) + c(dh - eg) to find the final value of |U|.
  • Calculate the determinants of the 2x2 matrices formed from the elements of U.
  • Solve the equation and find the value of |U|.

26. Matrix and Determinant - Problem on properties of determinants

• Problem 15
  • Given a matrix V = [[-3,1,2], [2,0,1], [1,3,-2]], find the value of |V|.
  • Apply the properties of determinants to simplify the calculation.
  • Use the formula |V| = a(ei - fh) - b(di - fg) + c(dh - eg) to find the final value of |V|.
  • Calculate the determinants of the 2x2 matrices formed from the elements of V.
  • Solve the equation and find the value of |V|.

27. Matrix and Determinant - Problem on properties of determinants

• Problem 16
  • Given a matrix W = [[-2,1,3], [1,0,-3], [2,1,0]], find the value of |W|.
  • Apply the properties of determinants to simplify the calculation.
  • Use the formula |W| = a(ei - fh) - b(di - fg) + c(dh - eg) to find the final value of |W|.
  • Calculate the determinants of the 2x2 matrices formed from the elements of W.
  • Solve the equation and find the value of |W|.

28. Matrix and Determinant - Problem on properties of determinants

• Problem 17
  • Given a matrix X = [[1,2,0,1], [-1,3,2,0], [2,1,2,2], [3,-2,0,1]], find the value of |X|.
  • Apply the properties of determinants to simplify the calculation.
  • Use the formula |X| = a(ei - fh) - b(di - fg) + c(dh - eg) to find the final value of |X|.
  • Calculate the determinants of the 3x3 matrices formed from the elements of X.
  • Solve the equation and find the value of |X|.

29. Matrix and Determinant - Problem on properties of determinants

• Problem 18
  • Given a matrix Y = [[1,0,2,3], [2,-1,3,2], [3,2,-1,4], [4,3,0,1]], find the value of |Y|.
  • Apply the properties of determinants to simplify the calculation.
  • Use the formula |Y| = a(ei - fh) - b(di - fg) + c(dh - eg) to find the final value of |Y|.
  • Calculate the determinants of the 3x3 matrices formed from the elements of Y.
  • Solve the equation and find the value of |Y|.

30. Matrix and Determinant - Problem on properties of determinants

• Problem 19
  • Given a matrix Z = [[2,1,-3,4], [-6,5,3,-2], [1,2,0,-3], [5,3,-2,1]], find the value of |Z|.
  • Apply the properties of determinants to simplify the calculation.
  • Use the formula |Z| = a(ei - fh) - b(di - fg) + c(dh - eg) to find the final value of |Z|.
  • Calculate the determinants of the 3x3 matrices formed from the elements of Z.
  • Solve the equation and find the value of |Z|.