Property 6 states that if any row or column of a matrix is multiplied by a constant, the determinant is multiplied by the same factor. Notice that the second and third columns are identical. According to Property 3, the determinant will be zero, so there is either no solution or an infinite number of solutions. We have to perform elimination to find out. Skip to main content. Search for:. Let M ij denote the i,j -th minor of A.
The coefficient of a nn in A is. Example 1. The eigen-values constitute the numbers l that make det l I-A zero.
Note that if any of b or c is 0, the only eigenvalue of A is a algebraically repeated n times. By a continuity argument the eigenvalues continue to be given by the above expression in all exceptional cases too.
Prove that a lower upper triangular matrix has an inverse iff none of its diagonal entries is zero and, moreover, the inverse of such a matrix is also lower upper triangular. Is the connverse true? Are such P and Q unique in general?
When would they be unique? Prove that a matrix which is strictly diagonally dominant with respect to either rows or columns is non-singular. If A is hermitian, has positive diagonal elements and is strictly diagonally dominant with respect to rows, show that A must be positive definite. Note that the span of an ordered set of vectors remains unchanged if elementary operations corresponding to eros viz.
The dimension of the span of a set of vectors is the maximal number of linearly independent vectors in the set. Hence, the elementary operations do not change the maximal number of linearly independent vectors in the set. The row rank rr A of a matrix A is the maximal number of linearly independent rows of the matrix. Similarly, the column rank cr A is the maximal number of linearly independent columns of A.
The row column space of a matrix being the span of its row column vectors has dimension rr A cr A. The rows of the highest order submatrix of non-zero determinant must be linearly independent else its determinant equals 0. Then all the more so are the same rows of the full matrix. Next, take a maximal collection of linearly independent rows and make a submatrix out of them. Row reduce it to its echelon form. Then each row of the echelon form must have a leading unity as the rows are linearly independent.
The submatrix of A consisting of these rows and the columns that contain leading unities of the echelon form is of a non-zero determinant. In view of the above theorem we define rank, r A , of the matrix to be any of the same rr A , cr A and dr A. Let the j-th column of B be denoted by B j. The result is immediate from this. Determinants are also used in calculus and linear algebra.
Determinants are defined only for square matrices. If the determinant of a matrix is 0, the matrix is said to be singular, and if the determinant is 1, the matrix is said to be unimodular. If the determinant is nonzero than there exists exactly one solution. If the determinant is zero, there could be no solutions, or there could be infinitely many.
For no solution, Two lines have no solution, if these two lines are parallel to each other. The lines are parallel to each other means that the slopes of the lines are equal. Sometimes that means that every single number is a solution, and sometimes it just means all the numbers that fit a certain pattern.
No solution would mean that there is no answer to the equation. It is impossible for the equation to be true no matter what value we assign to the variable. Infinite solutions would mean that any value for the variable would make the equation true. Note that we have variables on both sides of the equation.
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