# Prove that if A has rank r with r > 0, then A has an r×r invertible submatrix.

Question
Matrices
Prove that if A has rank r with r > 0, then A has an r×r invertible submatrix.

2021-02-15
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.

### Relevant Questions

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.
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 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}$$
If A and B are 3×3 invertible matrices, such that det(A)=2, det(B) =-2. Then det (ABA^T) = ???
Zero Divisors If a and b are real or complex numbers such thal ab = O. then either a = 0 or b = 0. Does this property hold for matrices? That is, if A and Bare n x n matrices such that AB = 0. is il true lhat we must have A = 0 or B = 0? Prove lhe resull or find a counterexample.
Determine whether A is similar B .If $$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}$$
that $$(BA^{-1})^T(A^{-1}B^T)^{-1} = I$$
Let A and B be similar nxn matrices. Prove that if A is idempotent, then B is idempotent. (X is idempotent if $$X^2=X$$ )