Mdhiggenz said:Homework Statement
Given A=[1 2 1; 0 4 3; 1 2 2]
determine the (2,3) entry of A-1 by computing a quotient of two determinants.
This problem confused me a bit, do they just want us to divide the adj(A) by the det(A) in order which would give us A-1 and just state the (2,3) entry from there?
Thanks
Higgenz
Homework Equations
The Attempt at a Solution
The linear algebra adjoint, also known as the adjugate or classical adjoint, is a matrix operation that is used to find the inverse of a square matrix. It is denoted as adj(A) or A*, and its elements are the determinants of the minors of the original matrix A.
To calculate the adjoint of a matrix, you need to first find the determinants of the minors of the original matrix. Then, you need to take the transpose of the matrix of minors and multiply it by -1 raised to the power of the sum of the row and column indices of each element. This will give you the adjoint matrix.
The determinant of a square matrix A can be calculated using the formula det(A) = A11*adj(A11) + A12*adj(A12) + ... + A1n*adj(A1n), where adj(Aij) represents the adjoint of the minor of A obtained by removing the i-th row and j-th column. In other words, the determinant of a matrix can be expressed in terms of its adjoints.
The adjoint matrix is important because it is used to find the inverse of a square matrix. It is also used in the Cramer's rule for solving systems of linear equations, and in the calculation of the eigenvalues and eigenvectors of a matrix.
No, the adjoint can only be calculated for square matrices. This is because the adjoint operation involves finding the determinants of minors, which can only be done for square matrices. For non-square matrices, there are other operations such as the pseudoinverse that can be used to find an "inverse-like" matrix.