Matrix and Determinant - Problems on Skew Symmetric and Symmetric matrices

  • In this lecture, we will solve some problems related to Skew Symmetric and Symmetric matrices.
  • These problems will help you understand the concepts better and prepare for the 12th Boards exam.
  • Let’s begin with a brief recap of Skew Symmetric and Symmetric matrices.
  • A matrix is said to be Skew Symmetric if its transpose is equal to the negative of the matrix itself.
  • A matrix is said to be Symmetric if its transpose is equal to the matrix itself.

Problem 1:

Find the values of ’m’ and ’n’ such that the matrix A = [[0, m], [-m, n]] is Skew Symmetric. Solution:

  1. The transpose of matrix A is given by A^T = [[0, -m], [m, n]].
  1. For A to be Skew Symmetric, A^T = -A.
  1. Equating the corresponding elements, we have:
    • 0 = 0
    • -m = m
    • m = -m
    • n = 0
  1. From the above equations, we can see that m = 0 and n = 0.
  1. Therefore, the matrix A = [[0, 0], [0, 0]] is Skew Symmetric.

Problem 2:

Given matrix B = [[3, 4], [4, k]]. Find the value of ‘k’ such that B is Symmetric. Solution:

  1. The transpose of matrix B is given by B^T = [[3, 4], [4, k]].
  1. For B to be Symmetric, B^T = B.
  1. Equating the corresponding elements, we have:
    • 3 = 3
    • 4 = 4
    • 4 = k
  1. From the above equations, we can see that k = 4.
  1. Therefore, the matrix B = [[3, 4], [4, 4]] is Symmetric.

Problem 3:

Find the product of matrix C = [[2, -1, 3], [0, 5, 2]] and its transpose. Solution:

  1. The transpose of matrix C is given by C^T = [[2, 0], [-1, 5], [3, 2]].
  1. To find the product of C and its transpose, we multiply their corresponding elements and sum them up.
  1. The product of C and C^T is given by:
    • [[2, -1, 3], [0, 5, 2]] x [[2, 0], [-1, 5], [3, 2]]
    • = [[4 + 1 + 9, 0 - 5 + 6], [0 + (-5) + 6, 0 + 25 + 4]]
    • = [[14, 1], [1, 29]]
  1. Therefore, the product of matrix C and its transpose is [[14, 1], [1, 29]].

Problem 4:

Find the determinant of matrix D = [[2, 3], [4, 5]]. Show all the steps. Solution:

  1. The determinant of a 2x2 matrix is given by the formula: ad - bc, where the matrix is [[a, b], [c, d]].
  1. For matrix D = [[2, 3], [4, 5]], the determinant can be calculated as:
    • det(D) = (2 * 5) - (3 * 4)
    • = 10 - 12
    • = -2
  1. Therefore, the determinant of matrix D is -2.

Problem 5:

Find the inverse of matrix E = [[2, 1], [3, 4]]. Show all the steps. Solution:

  1. To find the inverse of a 2x2 matrix, we can use the formula:
    • inverse(E) = (1/det(E)) x adj(E), where adj(E) is the adjugate of matrix E and det(E) is its determinant.
  1. The determinant of matrix E is det(E) = (2 * 4) - (1 * 3) = 8 - 3 = 5.
  1. The adjugate of matrix E is adj(E) = [[4, -1], [-3, 2]].
  1. Therefore, the inverse of matrix E can be calculated as:
    • inverse(E) = (1/5) x [[4, -1], [-3, 2]]
    • = [[4/5, -1/5], [-3/5, 2/5]].

That’s all for the first 10 slides. We will continue solving more problems on Skew Symmetric and Symmetric matrices in the next set of slides.

Matrix and Determinant - Problems on Skew Symmetric and Symmetric matrices

Slide 11

  • In the previous set of problems, we discussed the concepts of Skew Symmetric and Symmetric matrices.
  • Now, let’s move on to some more problems to reinforce our understanding.
  • These problems will cover various aspects of matrix multiplication and properties of determinants.
  • Let’s get started!

Problem 6:

Find the inverse of matrix F = [[1, 2, -1], [3, 4, 0], [2, 1, 2]]. Solution:

  1. To find the inverse of a matrix, we can use the formula:
    • inverse(F) = (1/det(F)) x adj(F), where adj(F) is the adjugate of matrix F and det(F) is its determinant.
  1. The determinant of matrix F can be calculated using cofactor expansion along the first row as :
    • det(F) = 1 * (42 - 01) - 2 * (32 - 01) + (-1) * (31 - 42)
    • = 1 * 8 - 2 * 6 + (-1) * (-5)
    • = 8 - 12 + 5
    • = 1
  1. The adjugate of matrix F can be obtained by taking the transpose of the matrix of cofactors:
    • adj(F) = [[4, -2, 6], [-3, 1, -8], [-2, 2, 4]]
  1. Therefore, the inverse of matrix F is given by:
    • inverse(F) = (1/1) x [[4, -2, 6], [-3, 1, -8], [-2, 2, 4]]
    • = [[4, -2, 6], [-3, 1, -8], [-2, 2, 4]]

Problem 7:

Find the product of matrix G = [[1, -2], [-3, 4]] and its inverse. Solution:

  1. The inverse of matrix G is given by:
    • inverse(G) = [[1/7, 2/7], [3/7, 1/7]]
  1. To find the product of G and its inverse, we multiply their corresponding elements and sum them up.
  1. The product of G and inverse(G) is given by:
    • [[1, -2], [-3, 4]] x [[1/7, 2/7], [3/7, 1/7]]
    • = [[(11) + (-23), (12) + (-21)], [(-31) + (43), (-32) + (41)]]
    • = [[-5, 0], [0, -5]]
  1. Therefore, the product of matrix G and its inverse is [[-5, 0], [0, -5]].

Problem 8:

Find the determinant of matrix H = [[1, 0], [0, 5], [2, 3]]. Solution:

  1. The given matrix H is a 3x2 matrix. The determinant of a non-square matrix is not defined.
  1. Therefore, the determinant of matrix H is not defined.

Problem 9:

Find the value of ‘k’ such that the matrix I = [[1, -2], [-4, k]] is Skew Symmetric. Solution:

  1. The transpose of matrix I is given by I^T = [[1, -4], [-2, k]].
  1. For I to be Skew Symmetric, I^T = -I.
  1. Equating the corresponding elements, we have:
    • 1 = -1
    • -4 = 2
    • -2 = -2
    • k = -4
  1. From the above equations, we can see that k = -4.
  1. Therefore, the matrix I = [[1, -2], [-4, -4]] is Skew Symmetric.

Problem 10:

Find the value of ‘p’ such that the matrix J = [[1, -p], [3, 2]] is Symmetric. Solution:

  1. The transpose of matrix J is given by J^T = [[1, 3], [-p, 2]].
  1. For J to be Symmetric, J^T = J.
  1. Equating the corresponding elements, we have:
    • 1 = 1
    • 3 = 3
    • -p = p
    • 2 = 2
  1. From the above equations, we can see that p can have any real value.
  1. Therefore, the matrix J = [[1, -p], [3, 2]] is Symmetric for any value of ‘p’.

That’s all for this set of slides. We will continue with more interesting problems on Skew Symmetric and Symmetric matrices in the next set.

Matrix and Determinant - Problems on Skew Symmetric and Symmetric matrices

Slide 21

  • In the previous set of problems, we discussed the concepts of Skew Symmetric and Symmetric matrices.
  • Now, let’s move on to some more problems to reinforce our understanding.
  • These problems will cover various aspects of matrix multiplication and properties of determinants.
  • Let’s get started!

Problem 11:

Find the inverse of matrix K = [[2, 1, -1], [3, -2, 4], [-6, 5, -3]]. Solution:

  1. To find the inverse of a matrix, we can use the formula:
    • inverse(K) = (1/det(K)) x adj(K), where adj(K) is the adjugate of matrix K and det(K) is its determinant.
  1. The determinant of matrix K can be calculated using cofactor expansion along the first row as :
    • det(K) = 2 * (-2* -3 - 45) - 1 * (3 -3 - 4* -6) - (-1) * (35 - -2 -6)
    • = 2 * (-6 + 20) - (-3 + 24) - (15 - 12)
    • = 2 * 14 - 21 - 3
    • = 28 - 21 - 3
    • = 4
  1. The adjugate of matrix K can be obtained by taking the transpose of the matrix of cofactors:
    • adj(K) = [[-14, -24, -20], [3, -6, 9], [-15, -12, -6]]
  1. Therefore, the inverse of matrix K is given by:
    • inverse(K) = (1/4) x [[-14, -24, -20], [3, -6, 9], [-15, -12, -6]]

Problem 12:

Find the product of matrix L = [[2, 3], [4, 5]] and its inverse. Solution:

  1. The inverse of matrix L is given by:
    • inverse(L) = [[-5, 3], [4, -2]]
  1. To find the product of L and its inverse, we multiply their corresponding elements and sum them up.
  1. The product of L and inverse(L) is given by:
    • [[2, 3], [4, 5]] x [[-5, 3], [4, -2]]
    • = [[(2*-5) + (34), (23) + (3*-2)], [(4*-5) + (54), (43) + (5*-2)]]
    • = [[-10 + 12, 6 - 6], [-20 + 20, 12 - 10]]
    • = [[2, 0], [0, 2]]
  1. Therefore, the product of matrix L and its inverse is [[2, 0], [0, 2]].

Problem 13:

Find the determinant of matrix M = [[1, 2, 3], [4, -5, 6], [-7, 8, -9]]. Solution:

  1. The given matrix M is a 3x3 matrix. The determinant of a 3x3 matrix can be calculated using cofactor expansion along the first row or the first column.
  1. Let’s calculate the determinant using cofactor expansion along the first column:
    • det(M) = 1 * (-5*(-9) - 68) - 2 * (4(-9) - 6*(-7)) + 3 * (48 - (-5)(-7))
    • = 1 * (45 - 48) - 2 * (-36 + 42) + 3 * (32 - 35)
    • = 1 * (-3) - 2 * 6 + 3 * (-3)
    • = -3 - 12 - 9
    • = -24
  1. Therefore, the determinant of matrix M is -24.

Problem 14:

Find the value of ‘k’ such that the matrix N = [[3, 2], [2, k]] is Skew Symmetric. Solution:

  1. The transpose of matrix N is given by N^T = [[3, 2], [2, k]].
  1. For N to be Skew Symmetric, N^T = -N.
  1. Equating the corresponding elements, we have:
    • 3 = -3
    • 2 = -2
    • 2 = 2
    • k = -k
  1. From the above equations, we can see that k = 0.
  1. Therefore, the matrix N = [[3, 2], [2, 0]] is Skew Symmetric.

Problem 15:

Find the value of ’m’ such that the matrix O = [[1, m], [m, 2]] is Symmetric. Solution:

  1. The transpose of matrix O is given by O^T = [[1, m], [m, 2]].
  1. For O to be Symmetric, O^T = O.
  1. Equating the corresponding elements, we have:
    • 1 = 1
    • m = m
    • m = m
    • 2 = 2
  1. From the above equations, we can see that m can have any real value.
  1. Therefore, the matrix O = [[1, m], [m, 2]] is Symmetric for any value of ’m’.

That’s all for this set of slides. We have covered diverse problems related to Skew Symmetric and Symmetric matrices. Keep practicing and you’ll excel in the topic for your 12th Boards exam!