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What is the order of the identity matrix In, when the matrix A is of order m x n?
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?
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?
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?
What is the result of multiplying a square matrix with its inverse?
What is the condition for computing the inverse of a matrix?
What is the condition for computing the inverse of a matrix?
What is the property of matrix algebra that states A(B+C) = AB + AC?
What is the property of matrix algebra that states A(B+C) = AB + AC?
What is the condition for a matrix to have an inverse?
What is the condition for a matrix to have an inverse?
How is the inverse of a diagonal matrix obtained?
How is the inverse of a diagonal matrix obtained?
What is the purpose of the adjoint in finding the inverse of a matrix?
What is the purpose of the adjoint in finding the inverse of a matrix?
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?
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?
What is a singular matrix?
What is a singular matrix?
What is the result of the operation AB, when A = [1 2; 3 4] and B = [1 0; 2 -3]?
What is the result of the operation AB, when A = [1 2; 3 4] and B = [1 0; 2 -3]?
What is the identity matrix used for in matrix inversion?
What is the identity matrix used for in matrix inversion?
How do you verify that you have correctly calculated an inverse matrix?
How do you verify that you have correctly calculated an inverse matrix?
What are the two main methods for computing the inverse of larger matrices?
What are the two main methods for computing the inverse of larger matrices?
What is the relationship between the adjoint of a matrix and its cofactors?
What is the relationship between the adjoint of a matrix and its cofactors?
What is the property of matrix algebra that states ImA = A = AIn, for any matrix A of order m x n?
What is the property of matrix algebra that states ImA = A = AIn, for any matrix A of order m x n?
Can you perform division operation on matrix?
Can you perform division operation on matrix?
What is the result of multiplying a matrix with its inverse?
What is the result of multiplying a matrix with its inverse?
What is the result of the operation BC, when B = [1 0; 2 -3] and C = [1 -1; 0 1]?
What is the result of the operation BC, when B = [1 0; 2 -3] and C = [1 -1; 0 1]?
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]?
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]?
What is the condition for a matrix to be invertible?
What is the condition for a matrix to be invertible?
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]?
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]?
What is the purpose of using the inversion technique in matrix equations?
What is the purpose of using the inversion technique in matrix equations?
What is the identity matrix of order 2 x 2?
What is the identity matrix of order 2 x 2?
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]?
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]?
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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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