Math Problems Vault

Problem 16.1

Give a description of all matrices of rank 0.

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Problem 16.2

Give a description of all matrices of rank 1.

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Problem 16.3

Is it possible for a matrix to have no basis minor?

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Problem 16.4

Specify some basis minor and determine the rank of the matrix:

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(1)

$\left[\begin{smallmatrix} 0 & 0 \\ 0 & 0\end{smallmatrix}\right]$

(2)

$\left[\begin{smallmatrix} 1 & - 1 \\ 2 & - 2\end{smallmatrix}\right]$

(3)

$\left[\begin{smallmatrix} 1 & 0 \\ 0 & 1\end{smallmatrix}\right]$

(4)

$\left[\begin{smallmatrix} 0 & 1 \\ 0 & 0\end{smallmatrix}\right]$

(5)

$\left[\begin{smallmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{smallmatrix}\right]$

(6)

$\left[\begin{smallmatrix} 1 & 1 & 1 \\ 2 & 2 & 3 \\ 3 & 3 & 4\end{smallmatrix}\right]$

(7)

$\left[\begin{smallmatrix} 2 & 1 & 1 & 1 & 1 \\ 2 & 1 & 1 & 2 & 3 \\ 4 & 2 & 2 & 3 & 4 \\ 2 & 1 & 1 & 1 & 1\end{smallmatrix}\right]$ .

Problem 16.5

Specify the basis rows:

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(1)

$\left[\begin{smallmatrix} 0 & 0 \\ 0 & 0\end{smallmatrix}\right]$

(2)

$\left[\begin{smallmatrix} 1 & - 1 \\ 2 & - 2\end{smallmatrix}\right]$

(3)

$\left[\begin{smallmatrix} 1 & 0 \\ 0 & 1\end{smallmatrix}\right]$

(4)

$\left[\begin{smallmatrix} 0 & 1 \\ 0 & 0\end{smallmatrix}\right]$

(5)

$\left[\begin{smallmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{smallmatrix}\right]$

(6)

$\left[\begin{smallmatrix} 1 & 1 & 1 \\ 2 & 2 & 3 \\ 3 & 3 & 4\end{smallmatrix}\right]$

(7)

$\left[\begin{smallmatrix} 2 & 1 & 1 & 1 & 1 \\ 2 & 1 & 1 & 2 & 3 \\ 4 & 2 & 2 & 3 & 4 \\ 2 & 1 & 1 & 1 & 1\end{smallmatrix}\right]$ .

Problem 16.6

Specify the basis columns:

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(1)

$\left[\begin{smallmatrix} 0 & 0 \\ 0 & 0\end{smallmatrix}\right]$

(2)

$\left[\begin{smallmatrix} 1 & - 1 \\ 2 & - 2\end{smallmatrix}\right]$

(3)

$\left[\begin{smallmatrix} 1 & 0 \\ 0 & 1\end{smallmatrix}\right]$

(4)

$\left[\begin{smallmatrix} 0 & 1 \\ 0 & 0\end{smallmatrix}\right]$

(5)

$\left[\begin{smallmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{smallmatrix}\right]$

(6)

$\left[\begin{smallmatrix} 1 & 1 & 1 \\ 2 & 2 & 3 \\ 3 & 3 & 4\end{smallmatrix}\right]$

(7)

$\left[\begin{smallmatrix} 2 & 1 & 1 & 1 & 1 \\ 2 & 1 & 1 & 2 & 3 \\ 4 & 2 & 2 & 3 & 4 \\ 2 & 1 & 1 & 1 & 1\end{smallmatrix}\right]$ .

Problem 16.7

Specify a basis minor, basis columns, and basis rows in a square matrix with determinant different from 0. What is the rank of such a matrix?

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Problem 16.8

Prove: the rank of a diagonal matrix equals the number of its entries that are nonzero.

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Problem 16.9

Prove: if all minors of order $k$ in a matrix are zero, then the minors of order $k + 1$ are also zero.

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Problem 16.10

Prove: the rank of a matrix is not less than the rank of any of its submatrices.

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Problem 16.11

Prove: appending a zero column to a matrix does not change its rank.

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Problem 16.12

Prove: appending to a matrix a column equal to a linear combination of its columns does not change its rank.

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Problem 16.13

Prove: if the columns of a matrix $B$ are linear combinations of the columns of a matrix $A$ , then $\operatorname {rg} B\leqslant \operatorname {rg} A$ .

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Problem 16.14

Estimate the rank of the matrix $\left\| AB\right\|^{\square }$ in terms of the ranks of the matrices $A$ and $B$ .

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Problem 16.15

Let the matrices $A$ and $B$ have the same height, and let the rank of $A$ not change after appending to it any of the columns of $B$ . Prove that $\operatorname {rg}\left\| AB\right\|^{\square } = \operatorname {rg} A$ .

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Problem 16.16

(1)(2)(3)(4)

Prove the following properties of the rank of a matrix:

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(1)

Multiplying some row of a matrix by a number different from zero does not change its rank.

(2)

Permuting the rows of a matrix does not change its rank.

(3)

Adding to some row of a matrix a linear combination of the remaining rows does not change its rank.

(4)

The rank of a matrix does not change under elementary transformations of its columns.

Problem 16.17

Describe a method for computing the rank of a matrix using elementary transformations of its rows and columns.

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Problem 16.18

Compute the rank of the matrix:

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(1)

$\left[\begin{smallmatrix} 1 & 0 \end{smallmatrix}\right]$

(2)

$\left[\begin{smallmatrix} 0 & 1 & 0 \end{smallmatrix}\right]$

(3)

$\mathbf{A}_{21}$ . $\mathbf{A}_{21} = \left[\begin{smallmatrix} 0 & 0 \\ 0 & 2 \end{smallmatrix}\right]$ .

(4)

$\mathbf{A}_{20}$ . $\mathbf{A}_{20} = \left[\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right]$ .

(5)

$\mathbf{A}_{13}$ . $\mathbf{A}_{13} = \left[\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right]$ .

(6)

$\mathbf{A}_{12}$ . $\mathbf{A}_{12} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right]$ .

(7)

$\mathbf{A}_{7}$ . $\mathbf{A}_{7} = \left[\begin{smallmatrix} 1+\sqrt{2} & 2-\sqrt{5} \\ 2+\sqrt{5} & 1-\sqrt{2} \end{smallmatrix}\right]$ .

(8)

$\mathbf{A}_{81}$ . $\mathbf{A}_{81} = \left[\begin{smallmatrix} 1+i \sqrt{2} & 3 \\ 1 & 1-i \sqrt{2} \end{smallmatrix}\right]$ .

(9)

$\mathbf{A}_{111}$ . $\mathbf{A}_{111} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \\ 2 & 2 \end{smallmatrix}\right]$ .

(10)

$\mathbf{A}_{202}$ . $\mathbf{A}_{202} = \left[\begin{smallmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ 2 & -2 & 1 \end{smallmatrix}\right]$ .

(11)

$\mathbf{A}_{205}$ . $\mathbf{A}_{205} = \left[\begin{smallmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{smallmatrix}\right]$ .

(12)

$\mathbf{A}_{233}$ . $\mathbf{A}_{233} = \left[\begin{smallmatrix} 1 & -1 & 2 \\ 3 & -3 & 6 \\ 2 & -2 & 4 \end{smallmatrix}\right]$ .

(13)

$\mathbf{A}_{214}$ . $\mathbf{A}_{214} = \left[\begin{smallmatrix} 13 & 16 & 16 \\ -5 & -7 & -6 \\ -6 & -8 & -7 \end{smallmatrix}\right]$ .

(14)

$\mathbf{A}_{232}$ . $\mathbf{A}_{232} = \left[\begin{smallmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 1 & 1 & 1 \end{smallmatrix}\right]$ .

(15)

$\mathbf{A}_{368}$ . $\mathbf{A}_{368} = \left[\begin{smallmatrix} 1+i \sqrt{2} & i-\sqrt{2} & 1 \\ 1+i \sqrt{3} & i-\sqrt{3} & 1 \\ 1+i \sqrt{4} & i-\sqrt{4} & 1 \end{smallmatrix}\right]$ .

(16)

$\mathbf{A}_{396}$ . $\mathbf{A}_{396} = \left[\begin{smallmatrix} 1 & 1 & 3 \\ 2 & 2 & 6 \\ 0 & 3 & 0 \\ 0 & 4 & 0 \end{smallmatrix}\right]$ .

(17)

$\mathbf{A}_{408}$ . $\mathbf{A}_{408} = \left[\begin{smallmatrix} 2 & 4 & 2 \\ -1 & -2 & -1 \\ 1 & 5 & 3 \\ 8 & 1 & -2 \\ 2 & 7 & 4 \end{smallmatrix}\right]$ .

(18)

$\mathbf{A}_{452}$ . $\mathbf{A}_{452} = \left[\begin{smallmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 3 \\ 1 & 2 & 3 & 4 \end{smallmatrix}\right]$ .

(19)

$\mathbf{A}_{435}$ . $\mathbf{A}_{435} = \left[\begin{smallmatrix} 0 & 0 & 9 & 1 \\ 0 & 0 & 9 & 2 \\ 0 & 0 & 9 & 3 \\ 1 & 2 & 3 & 4 \end{smallmatrix}\right]$ .

(20)

$\mathbf{A}_{453}$ . $\mathbf{A}_{453} = \left[\begin{smallmatrix} 0 & 0 & 1 & 4 \\ 0 & 0 & 2 & 5 \\ 1 & 2 & 3 & 6 \\ 4 & 5 & 6 & 7 \end{smallmatrix}\right]$ .

(21)

$\mathbf{A}_{444}$ . $\mathbf{A}_{444} = \left[\begin{smallmatrix} 5 & 4 & 1 & 3 \\ 2 & 1 & 1 & 4 \\ 3 & 2 & 1 & 1 \\ 1 & 3 & -2 & 2 \end{smallmatrix}\right]$ .

(22)

$\mathbf{A}_{454}$ . $\mathbf{A}_{454} = \left[\begin{smallmatrix} 2 & 4 & 6 & 1 \\ 1 & 2 & 3 & 1 \\ -3 & -6 & -9 & 1 \\ 1 & 2 & 3 & 1 \end{smallmatrix}\right]$ .

(23)

$\mathbf{A}_{443}$ . $\mathbf{A}_{443} = \left[\begin{smallmatrix} 25 & 31 & 17 & 43 \\ 75 & 94 & 53 & 132 \\ 75 & 94 & 54 & 134 \\ 25 & 32 & 20 & 48 \end{smallmatrix}\right]$ .

(24)

$\mathbf{A}_{587}$ . $\mathbf{A}_{587} = \left[\begin{smallmatrix} 1 & 2 & 2 & 5 & 3 \\ 2 & 3 & -5 & 26 & -4 \\ 3 & -4 & 8 & -9 & 1 \\ -4 & 1 & -3 & -12 & 2 \end{smallmatrix}\right]$ .

(25)

$\mathbf{A}_{533}$ . $\mathbf{A}_{533} = \left[\begin{smallmatrix} 1 & 10 & 100 & 10^{3} & 10^{4} \\ 0,1 & 2 & 30 & 400 & 5000 \\ 0 & 0,1 & 3 & 60 & 800 \\ 0 & 0 & 0,1 & 4 & 90 \\ 0 & 0 & 0 & 0,1 & 5 \end{smallmatrix}\right]$ .

(26)

$\mathbf{A}_{544}$ . $\mathbf{A}_{544} = \left[\begin{smallmatrix} 1 & 2 & 3 & 0 & 0 \\ 2 & 3 & 4 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 3 & 4 \end{smallmatrix}\right]$ .

(27)

$\mathbf{A}_{592}$ . $\mathbf{A}_{592} = \left[\begin{smallmatrix} 1 & -1 & -1 & 1 & 0 & 0 \\ -1 & 2 & 4 & 0 & 1 & 5 \\ 2 & -3 & -5 & 1 & 0 & -4 \\ 0 & 1 & 3 & 1 & 1 & 5 \\ 0 & 1 & 3 & 1 & 0 & 4 \\ 1 & 0 & 2 & 2 & 1 & 5 \end{smallmatrix}\right]$ .

(28)

$\mathbf{A}_{632}$ . $\mathbf{A}_{632} = \left[\begin{smallmatrix} 0 & 1 & 1 & \ldots & 1 & 1 & 1 \\ -1 & 0 & 1 & \ldots & 1 & 1 & 1 \\ -1 & -1 & 0 & \ldots & 1 & 1 & 1 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ -1 & -1 & -1 & \ldots & -1 & -1 & 0 \end{smallmatrix}\right]$ .

(29)

$\mathbf{A}_{634}$ . $\mathbf{A}_{634} = \left[\begin{smallmatrix} 0 & 1 & 0 & 0 & \ldots & 0 & 0 \\ 1 & 0 & 1 & 0 & \ldots & 0 & 0 \\ 0 & 1 & 0 & 1 & \ldots & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & 0 & \ldots & 0 & 1 \\ 0 & 0 & 0 & 0 & \ldots & 1 & 0 \end{smallmatrix}\right]$ .

Problem 16.19

Compute the rank of the matrix for all possible values of the parameter:

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(1)

$\mathbf{A}_{78}$ . $\mathbf{A}_{78} = \left[\begin{smallmatrix} \varepsilon & 1 \\ -1 & \varepsilon \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{367}$ . $\mathbf{A}_{367} = \left[\begin{smallmatrix} 3 & 6 & 9 \\ 4 & 8 & 12 \\ 2 & 7 & \lambda \end{smallmatrix}\right]$ .

(3)

$\mathbf{A}_{365}$ . $\mathbf{A}_{365} = \left[\begin{smallmatrix} 1 & 1 & 1 \\ 1 & \alpha & \alpha \\ 1 & \alpha^{2} & \alpha^{2} \end{smallmatrix}\right]$ .

(4)

$\mathbf{A}_{363}$ . $\mathbf{A}_{363} = \left[\begin{smallmatrix} 1 & 1 & 1 \\ 1 & \omega & \omega^{2} \\ 1 & \omega^{2} & \omega \end{smallmatrix}\right]$ .

(5)

$\mathbf{A}_{508}$ . $\mathbf{A}_{508} = \left[\begin{smallmatrix} 1 & \lambda & -1 & 2 \\ 2 & -1 & \lambda & 5 \\ 1 & 10 & -6 & 1 \end{smallmatrix}\right]$ .

(6)

$\mathbf{A}_{629}$ . $\mathbf{A}_{629} = \left[\begin{smallmatrix} 1-\lambda & 2 & 3 & \ldots & n \\ 1 & 2-\lambda & 3 & \ldots & n \\ \ldots \ldots & \ldots & \ldots & \ldots & \ldots \\ 1 & 2 & 3 & \ldots & n-\lambda \end{smallmatrix}\right]$ .

(7)

$\mathbf{A}_{645}$ . $\mathbf{A}_{645} = \left[\begin{smallmatrix} 1 & 1 & 1 & \ldots & 1 \\ 1 & \varepsilon & \varepsilon^2 & \ldots & \varepsilon^{n-1} \\ 1 & \varepsilon^2 & \varepsilon^4 & \ldots & \varepsilon^{2(n-1)} \\ \ldots \ldots \ldots \ldots & \ldots \ldots & \ldots \ldots & \ldots \ldots \\ 1 & \varepsilon^{n-1} & \varepsilon^{2(n-1)} & \ldots & \varepsilon^{(n-1)^2} \end{smallmatrix}\right]$ .

Problem 16.20

(1)(2)(3)

Compute the rank of the matrix $A - \lambda E$ for all values of the parameter $\lambda$ , if:

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(1)

$A = \mathbf{A}_{47}$ . $\mathbf{A}_{47} = \left[\begin{smallmatrix} 5 & -2 \\ -2 & 8 \end{smallmatrix}\right]$ .

(2)

$A = \mathbf{A}_{211}$ . $\mathbf{A}_{211} = \left[\begin{smallmatrix} 3 & 0 & 0 \\ 2 & 6 & 4 \\ -2 & -3 & -1 \end{smallmatrix}\right]$ .

(3)

$A = \mathbf{A}_{431}$ . $\mathbf{A}_{431} = \left[\begin{smallmatrix} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{smallmatrix}\right]$ .

Problem 16.21

Prove that if $\operatorname {det} A = 0$ , then the rows of the matrix $A$ , as well as its columns, are linearly dependent.

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Problem 16.22

The matrix $A$ has order $n$ and contains a zero submatrix of order $n - 1$ . Estimate the rank of $A$ .

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Problem 16.23

The matrix $A$ has order $n$ and contains a zero submatrix of order $s$ . Estimate the rank of $A$ .

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Problem 16.24

The matrix $A$ has order $n$ and contains a submatrix of order $n - 1$ having rank 1. Estimate the rank of $A$ .

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Problem 16.25

(1)(2)

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(1)

Estimate the rank of the product of two matrices in terms of the ranks of the factors.

(2)

Give examples in which the following relations hold: $\operatorname {rg} AB < \operatorname {rg} A, \quad \operatorname {rg} AB < \operatorname {rg} B, \quad \operatorname {rg} AB < \min (\operatorname {rg} A, \operatorname {rg} B), \quad \operatorname {rg} AB = \operatorname {rg} A, \quad \operatorname {rg} AB = \operatorname {rg} B$ .

Problem 16.26

(1)(2)

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(1)

Let $\mathbf{a}$ be a row, b a column. Compute the rank of the matrix ba

(2)

(р). Let $\operatorname {rg} A = 1$ . Prove that the matrix $A$ equals the product of some column and some row.

Problem 16.27

Let $A, B, C$ be matrices, $\operatorname {det} A \neq 0$ , and let the products $AB, CA$ be defined. Prove that $\operatorname {rg} AB = \operatorname {rg} B, \operatorname {rg} CA = \operatorname {rg} C$ . Can either of these equalities hold if $\operatorname {det} A = 0$ ?

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Problem 16.28

Prove that if $\operatorname {rg} A = r$ , then the minor standing at the intersection of $r$ linearly independent rows and $r$ linearly independent columns of the matrix $A$ is nonzero.

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Problem 16.29

Let the matrix $A$ consist of $r$ linearly independent columns, and $B$ of $r$ linearly independent rows. What is the rank of $AB$ ?

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Problem 16.30

The matrices $A$ and $B$ have sizes $m \times r$ and $r \times n$ respectively, and $\operatorname {rg} AB = r$ . Find the ranks of the matrices $A$ and $B$ .

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Problem 16.31

(1)(2)(3)

A factorization of the matrix $A$ into a product $A = BC$ is called a skeleton factorization if $\operatorname {rg} A = \operatorname {rg} B = \operatorname {rg} C$ and the matrices $B$ and $C$ have full rank (i.e., a rank equal to one of the dimensions of the matrix)

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(1)

Prove that any matrix $A$ can be represented as the product of a matrix $M$ , consisting of the basis rows of $A$ , and some matrix $K$ (a skeleton factorization of the matrix by rows).

(2)

State and prove the analogous statement for the skeleton factorization of a matrix by columns.

(3)

How are the various skeleton factorizations of one matrix related to each other?

Problem 16.32

Find some skeleton factorizations (see problem 16.31) by rows and by columns for the following matrices:

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(1)

$\mathbf{A}_{14}$ . $\mathbf{A}_{14} = \left[\begin{smallmatrix} 1 & 0 \\ 1 & 0 \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{231}$ . $\mathbf{A}_{231} = \left[\begin{smallmatrix} 8 & -12 & 0 \\ 6 & -9 & 0 \\ 2 & -3 & 0 \end{smallmatrix}\right]$ .

(3)

$\mathbf{A}_{251}$ . $\mathbf{A}_{251} = \left[\begin{smallmatrix} 0 & 0 & -1 \\ 0 & 0 & 4 \\ 1 & -4 & 0 \end{smallmatrix}\right]$ .

(4)

$\mathbf{A}_{403}$ . $\mathbf{A}_{403} = \left[\begin{smallmatrix} 2 & 3 & -1 \\ 4 & 7 & 5 \\ -1 & -2 & -3 \\ 2 & 5 & 13 \end{smallmatrix}\right]$ .

(5)

$\mathbf{A}_{454}$ . $\mathbf{A}_{454} = \left[\begin{smallmatrix} 2 & 4 & 6 & 1 \\ 1 & 2 & 3 & 1 \\ -3 & -6 & -9 & 1 \\ 1 & 2 & 3 & 1 \end{smallmatrix}\right]$ .

Problem 16.33

Prove that any matrix of rank $r$ can be represented as a sum of $r$ matrices of rank 1.

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Problem 16.34

Indicate which of the relations written below are possible. Which of them hold for an arbitrary pair of matrices of the same size?

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(1)

$\operatorname {rg}(A + B) = \operatorname {rg} A$

(2)

$\operatorname {rg}(A + B) = \max (\operatorname {rg} A, \operatorname {rg} B)$

(3)

$\operatorname {rg}(A + B) = \operatorname {rg} A + \operatorname {rg} B$

(4)

$\operatorname {rg}(A + B) < \min (\operatorname {rg} A, \operatorname {rg} B)$

(5)

$\operatorname {rg}(A + B) < \operatorname {rg} A + \operatorname {rg} B$

(6)

$\operatorname {rg}(A + B) \leqslant \operatorname {rg} A + \operatorname {rg} B$ .

Problem 16.35

Let the matrices $A$ and $B$ have sizes $m \times n$ and $n \times p$ respectively, and let $AB = O$ . Prove that $\operatorname {rg} A + \operatorname {rg} B \leqslant n$ .

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Problem 16.36

Prove that

\operatorname {rg}\left[\begin{smallmatrix} \sin (a_1 + b_1) & \sin (a_1 + b_2) & \cdots & \sin (a_1 + b_n) \\ \sin (a_2 + b_1) & \sin (a_2 + b_2) & \cdots & \sin (a_2 + b_n) \\ \vdots & \vdots & \vdots & \vdots \\ \sin (a_n + b_1) & \sin (a_n + b_2) & \cdots & \sin (a_n + b_n) \\ \end{smallmatrix}\right] \leqslant 2

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Problem 16.37

Prove that

\operatorname {rg}\left[\begin{smallmatrix} A & O \\ O & B\end{smallmatrix}\right]^{\square } = \operatorname {rg} A + \operatorname {rg} B.

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Problem 16.38

Prove that

\operatorname {rg}\left[\begin{smallmatrix} A & C \\ O & B\end{smallmatrix}\right]^{\square } \geqslant \operatorname {rg} A + \operatorname {rg} B.

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Problem 16.39

Let $A$ be a square matrix. Prove that

\operatorname {rg}\left[\begin{smallmatrix} A & A^{2} \\ A^{3} & A^{4} \end{smallmatrix}\right]^{\square } = \operatorname {rg} A.

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Problem 16.40

Let $E$ be the identity matrix, and let $A, B$ be arbitrary square matrices of order $n$ . Prove that

\mathrm{rg}\left[\begin{smallmatrix} E & B \\ A & AB\end{smallmatrix}\right]^{\square } = n.

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Problem 16.41

Prove that

\operatorname {rg}\left[\begin{smallmatrix} A & B \\ 3A & - B\end{smallmatrix}\right]^{\square } = \operatorname {rg} A + \operatorname {rg} B.

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Problem 16.42

Let $A$ be a nonsingular square matrix of order $n$ , and let the matrices $B, C$ and $D$ be rectangular. Find a necessary and sufficient condition for

\operatorname {rg}\left[\begin{smallmatrix} A & B \\ C & D\end{smallmatrix}\right]^{\square } = n.

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