Business Mathematics: Matrix Inverse
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Questions and Answers

What is the order of the identity matrix In, when the matrix A is of order m x n?

  • p x q
  • m x m
  • m x p
  • n x n (correct)

    • What is the definition of the inverse of a matrix?

  • A matrix obtained by multiplying the adjoint with determinant
  • A matrix obtained by dividing the adjoint with determinant (correct)
  • A matrix obtained by subtracting the adjoint with determinant
  • A matrix obtained by adding the adjoint with determinant

    • What is the requirement for the order of matrices A and B in the equation AX = B?

  • The order of A must be amenable to multiplication with the order of B (correct)
  • The order of A and B can be any
  • The order of A must be greater than the order of B
  • The order of A and B must be the same

    • What is the result of multiplying a square matrix with its inverse?

    <p>An identity matrix</p> Signup and view all the answers

    What is the condition for computing the inverse of a matrix?

    <p>The matrix is square with a non-zero determinant</p> Signup and view all the answers

    What is the property of matrix algebra that states A(B+C) = AB + AC?

    <p>Distributive property</p> Signup and view all the answers

    What is the condition for a matrix to have an inverse?

    <p>The matrix must be a square matrix</p> Signup and view all the answers

    How is the inverse of a diagonal matrix obtained?

    <p>By replacing each element in the diagonal with its reciprocal</p> Signup and view all the answers

    What is the purpose of the adjoint in finding the inverse of a matrix?

    <p>To divide the determinant to find the inverse</p> Signup and view all the answers

    What is the result of the operation AI, when A = [1 2; 3 4] and I is the identity matrix of order 2 x 2?

    <p>[1 2; 3 4]</p> Signup and view all the answers

    What is a singular matrix?

    <p>A matrix that does not have an inverse</p> Signup and view all the answers

    What is the result of the operation AB, when A = [1 2; 3 4] and B = [1 0; 2 -3]?

    <p>[5 -6; 11 -12]</p> Signup and view all the answers

    What is the identity matrix used for in matrix inversion?

    <p>To verify the result of matrix inversion</p> Signup and view all the answers

    How do you verify that you have correctly calculated an inverse matrix?

    <p>By multiplying the inverse with the original matrix</p> Signup and view all the answers

    What are the two main methods for computing the inverse of larger matrices?

    <p>Using Minors and Cofactors, and using Elementary Row Operations</p> Signup and view all the answers

    What is the relationship between the adjoint of a matrix and its cofactors?

    <p>The adjoint is the transpose of the cofactors</p> Signup and view all the answers

    What is the property of matrix algebra that states ImA = A = AIn, for any matrix A of order m x n?

    <p>Identity property</p> Signup and view all the answers

    Can you perform division operation on matrix?

    <p>No, matrix division is not defined</p> Signup and view all the answers

    What is the result of multiplying a matrix with its inverse?

    <p>An identity matrix</p> Signup and view all the answers

    What is the result of the operation BC, when B = [1 0; 2 -3] and C = [1 -1; 0 1]?

    <p>[1 -1; 2 -5]</p> Signup and view all the answers

    What is the result of the operation A(BC), when A = [1 2; 3 4], B = [1 0; 2 -3], and C = [1 -1; 0 1]?

    <p>[11 -12; 14 -23]</p> Signup and view all the answers

    What is the condition for a matrix to be invertible?

    <p>The matrix is square and has a non-zero determinant</p> Signup and view all the answers

    What is the result of the operation AB + AC, when A = [1 2; 3 4], B = [1 0; 2 -3], and C = [1 -1; 0 1]?

    <p>[11 -12; 14 -23]</p> Signup and view all the answers

    What is the purpose of using the inversion technique in matrix equations?

    <p>To solve for the unknown value of a matrix X</p> Signup and view all the answers

    What is the identity matrix of order 2 x 2?

    <p>[1 0; 0 1]</p> Signup and view all the answers

    What is the result of the operation (AB)C, when A = [1 2; 3 4], B = [1 0; 2 -3], and C = [1 -1; 0 1]?

    <p>[11 -12; 14 -23]</p> Signup and view all the answers

    Study Notes

    Matrix Inverse

    • Inversion of a matrix is a process of finding a matrix that, when multiplied by the original matrix, results in the identity matrix.
    • The inverse of a matrix is denoted by A^(-1).

    Properties of Matrix Inverse

    • The result of multiplying a square matrix with its inverse is the identity matrix (I).
    • The inverse of a matrix can only be applied to square matrices.
    • A singular matrix is a matrix that does not have an inverse.
    • An invertible matrix is a matrix that has an inverse.

    Identifying Invertible Matrices

    • A matrix is invertible if it is a square matrix and has a non-zero determinant.

    Adjoint of a Matrix

    • The adjoint of a matrix is the transpose of the cofactor matrix.
    • The adjoint is used to find the inverse of a matrix.

    Calculating Inverse of a Matrix

    • The inverse of a matrix can be calculated using the adjoint and determinant.
    • The formula for calculating the inverse is: A^(-1) = adj(A) / |A|.
    • The inverse of a diagonal matrix is obtained by replacing each element in the diagonal with its reciprocal.

    Matrix Operations

    • The associative property of matrix multiplication is: (AB)C = A(BC).
    • The distributive property of matrix multiplication is: A(B+C) = AB + AC.
    • The identity matrix for any matrix A of order m x n is: Im A = A = A In.
    • For a square matrix A of order n x n, In A = A In = A.

    Verifying Inverse of a Matrix

    • To verify if the inverse of a matrix is correct, multiply the original matrix with the calculated inverse matrix. The result should be the identity matrix.
    • Division operation cannot be performed on matrices. Instead, we use the inverse of a matrix to solve for unknown values.

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    Business Mathematics Part A PDF

    Description

    Assess your understanding of matrix inversion, including definitions, types of matrices, and calculations. Test your knowledge of inverse matrices and singular matrices.

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