It is given that rank of A is r. Then A has rr independent columns. Let B be the submatrix formed by those rr columns of A. Clearly the rank of B is r. We know that rank of a matrix is equal to dimension of the row space of that matrix. Therefore dimension of the row space of B also r. Therese rr rows are linearly independemmt. Form the submatrix B if we choose particularly those r rows, then is an r×r submatrix which is invertible.
Prove that if A has rank r with r > 0, then A has an r×r invertible submatrix.
Question
Answers (1)
2021-02-15
Relevant Questions
asked 2021-01-04
In the following question there are statements which are TRUE and statements which are FALSE.
Choose all the statements which are FALSE.
1. If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent - thus no solution.
2. If B has a column with zeros, then AB will also have a column with zeros, if this product is defined.
3. If AB + BA is defined, then A and B are square matrices of the same size/dimension/order.
4. Suppose A is an n x n matrix and assume A^2 = O, where O is the zero matrix. Then A = O.
5. If A and B are n x n matrices such that AB = I, then BA = I, where I is the identity matrix.
Choose all the statements which are FALSE.
1. If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent - thus no solution.
2. If B has a column with zeros, then AB will also have a column with zeros, if this product is defined.
3. If AB + BA is defined, then A and B are square matrices of the same size/dimension/order.
4. Suppose A is an n x n matrix and assume A^2 = O, where O is the zero matrix. Then A = O.
5. If A and B are n x n matrices such that AB = I, then BA = I, where I is the identity matrix.
asked 2020-11-24
Let \(A=I_2\) and \(B=\begin{bmatrix}1 & 1 \\0 & 1 \end{bmatrix}\) . Discuss the validity of the following statement and cite reasons for your conclusion:
If A and B have the same trace , determinant , rank , and eigenvalues , then the matrices are similar.
If A and B have the same trace , determinant , rank , and eigenvalues , then the matrices are similar.
asked 2020-12-15
If the product D=ABC of three square matrices is invertible , then A must be invertible (so are B and C). Find a formula for \(A^{-1} (i.e. A^{-1}=\dotsb) that involves only the matrices \(A, B B^{-1} , C, C^{-1} , D \text{ and/or } D^{-1}\)
asked 2020-11-07
If A and B are 3×3 invertible matrices, such that det(A)=2, det(B) =-2. Then det (ABA^T) = ???
asked 2021-03-02
Zero Divisors If a and b are real or complex numbers such thal ab = O. then either a = 0 or b = 0. Does this property hold for matrices? That is, if A and Bare n x n matrices such that AB = 0. is il true lhat we must have A = 0 or B = 0? Prove lhe resull or find a counterexample.
asked 2020-10-26
Determine whether A is similar B .If \(A \sim B\), give an invertible matrix P such that \(P^{-1}AP = B\)
\(A=\begin{bmatrix}1 & 1&0 \\0&1&1\\0&0&1 \end{bmatrix} , B=\begin{bmatrix}1&1&0 \\0&1&0\\0&0&1 \end{bmatrix}\)
\(A=\begin{bmatrix}1 & 1&0 \\0&1&1\\0&0&1 \end{bmatrix} , B=\begin{bmatrix}1&1&0 \\0&1&0\\0&0&1 \end{bmatrix}\)
asked 2020-11-11
Prove that if A and B are similar n x n matrices, then tr(A) = tr(B).
asked 2020-11-12
Prove that if A and B are n x n matrices, then tr(AB) = tr(BA).
asked 2020-12-30
If A,B are symmetric matrices, then prove
that \((BA^{-1})^T(A^{-1}B^T)^{-1} = I\)
that \((BA^{-1})^T(A^{-1}B^T)^{-1} = I\)
asked 2021-03-18
Let A and B be similar nxn matrices. Prove that if A is idempotent, then B is idempotent. (X is idempotent if \(X^2=X\) )
