Solving System of Linear Equations using Inverse of a Matrix
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Questions and Answers

What is the necessary and sufficient condition for a square matrix A to be invertible?

  • A is an identity matrix
  • A is a square matrix of any order
  • A is a nonsingular matrix (correct)
  • A is a diagonal matrix

    • If A and B are nonsingular matrices of the same order, which of the following statements is true?

  • AB is nonsingular, but BA is singular
  • AB and BA are both nonsingular matrices (correct)
  • AB is singular, but BA is nonsingular
  • AB and BA are both singular matrices

    • If A and B are square matrices of the same order, what is the relationship between the determinants of A, B, and their product AB?

  • |AB| = |A| + |B|
  • |AB| = |A| - |B|
  • |AB| = |A| * |B| (correct)
  • |AB| = |A| / |B|

    • What is the value of the determinant of the adjoint of a square matrix A of order n?

    <p>|adj(A)| = |A|^(n-1)</p> Signup and view all the answers

    If A is an invertible matrix, which of the following statements is true?

    <p>A(adj A) = (adj A)A = I</p> Signup and view all the answers

    If A is a $3 \times 3$ matrix and $|A| = 5$, what is the value of $|adj(A)|$?

    <p>25</p> Signup and view all the answers

    If A is a nonsingular matrix, what is the inverse of A, denoted by A^(-1)?

    <p>A^(-1) = adj(A)/|A|</p> Signup and view all the answers

    If A and B are nonsingular matrices of the same order, which of the following statements is true about their inverses?

    <p>(AB)^(-1) = A^(-1) * B^(-1)</p> Signup and view all the answers

    If A is a nonsingular matrix and I is the identity matrix of the same order, which of the following statements is true?

    <p>A * I = I</p> Signup and view all the answers

    If A is a $2 \times 2$ matrix with $|A| = 6$, what is the value of $|adj(A)|$?

    <p>6</p> Signup and view all the answers

    Study Notes

    Nonsingular Matrices

    • A matrix A is nonsingular if its determinant is not equal to zero.
    • If A and B are nonsingular matrices of the same order, then AB and BA are also nonsingular matrices of the same order.

    Determinant of Product of Matrices

    • The determinant of the product of matrices is equal to the product of their respective determinants.
    • If A is a square matrix of order n, then |adj(A)| = |A|^(n-1).

    Invertible Matrices

    • A square matrix A is invertible if and only if A is a nonsingular matrix.
    • If A is invertible, then A^(–1) = (1/|A|)adj(A).

    Solution of System of Linear Equations

    • The system of linear equations can be written as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
    • If A is a nonsingular matrix, then the system of equations has a unique solution given by X = A^(–1)B.
    • If A is a singular matrix, then the system of equations may have no solution or infinitely many solutions.

    Adjoint of a Matrix

    • The adjoint of a square matrix A is defined as the transpose of the matrix of cofactors.
    • Adjoint of the matrix A is denoted by adj(A).
    • adj(A) can be used to find the inverse of a matrix A, i.e., A^(–1) = (1/|A|)adj(A).

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    Description

    Learn how to solve a system of linear equations by expressing them as matrix equations and using the inverse of the coefficient matrix. Understand the process of finding the solution by matrix manipulation.

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