Question

A matrix obtained from \(A\) by deleting rows and columns is called a submatrix of \(A .\) If \(A\) has an invertible \(k \times k\) submatrix, show that rank \(A \geq k\). [Hint: Show that row and column operations carry \(A \rightarrow\left[\begin{array}{rr}I_{k} & P \\ 0 & Q\end{array}\right]\) in block form.] Remark: It can be shown that rank \(A\) is the largest integer \(r\) such that \(A\) has an invertible \(r \times r\) submatrix.

Short answer

If a matrix has an invertible \( k \times k \) submatrix, rank of the matrix is at least \( k \).

Step-by-step solution

Understand the Problem Statement

The problem requires demonstrating that if a matrix \( A \) has an invertible \( k \times k \) submatrix, then the rank of matrix \( A \) is at least \( k \). This means we need to connect the invertibility of a submatrix to the rank of the entire matrix.

Consider a Submatrix Initialization

Assume that the \( k \times k \) submatrix, \( B \), is formed by deleting specific rows and columns from \( A \), and \( B \) is invertible. This means \( \ ext {det}(B) eq 0 \), ensuring full rank of \( B \).

Apply Elementary Row and Column Operations

By applying a series of row and column operations (which do not change the rank of a matrix), transform the original matrix \( A \) to a new matrix of the form:\[\begin{bmatrix}I_k & P \0 & Q\end{bmatrix}.\]This form suggests the presence of an identity matrix \( I_k \) as a block indicating that there is a fully independent set of \( k \) columns.

Conclude Foaming of Block Matrices

In this transformed matrix, the presence of the \( k \times k \) identity matrix \( I_k \) ensures that there are at least \( k \) linearly independent rows and columns. Therefore, the rank of the matrix is at least \( k \).

Connect Submatrix and Rank

The invertibility of the submatrix means it is full-rank and can be rotated via operations that essentially showcase at least \( k \) independent directions or dimensions in \( A \). Thus, the rank \( A \geq k \).

Key concepts

Submatrix

A submatrix is formed by removing specific rows and columns from a given larger matrix to create a smaller matrix within it. These can be strategically chosen from the original matrix, resulting in a new matrix that highlights certain properties. It's often used when focusing on specific parts of data or characteristics.

For example, consider a matrix:

• Original matrix, let's say matrix \( A \)
• By removing particular rows and columns, we obtain a submatrix.

Submatrices play a critical role in linear algebra because they allow us to examine smaller, more manageable pieces of a matrix. If a submatrix is invertible (or 'non-singular'), it indicates that it might have important implications on the rank and structure of the original matrix.

Invertible Matrix

An invertible matrix, also known as a non-singular matrix, is a square matrix that has a matrix inverse. This means there is another matrix which, when multiplied with the original matrix, results in the identity matrix. Being invertible implies that:

• The determinant of the matrix is not zero \( \text{det}(A) eq 0 \).
• The matrix is of full rank, meaning all its rows and columns are linearly independent.

In this exercise, if the submatrix is invertible, it implies that this smaller matrix retains all the necessary ranks or independent variables, signifying important information about the original matrix.

Being able to invert a matrix means it's essentially 'solvable', allowing transformations that maintain crucial properties like rank.

Elementary Row Operations

Elementary row operations are key tools in matrix manipulations and solving linear algebraic equations. These include:

• Swapping two rows.
• Multiplying a row by a non-zero scalar.
• Adding or subtracting multiples of rows from one another.

These operations are essential because they enable reducing complex matrices into simpler forms, like row-echelon form or reduced row-echelon form, without altering the matrix's rank.

In the exercise, applying these operations reveals the structure and nature of the matrices involved by simplifying the matrix to a form that includes the identity matrix—demonstrating the independence of columns and, thereby, establishing a connection to rank.

Linear Independence

Linear independence is a fundamental concept in understanding matrices and their dimensions. A set of vectors is considered linearly independent if none of the vectors in the set can be written as a combination of others. This directly affects the rank of a matrix because:

• A matrix's rank is defined by the maximum number of linearly independent row or column vectors.
• A higher rank indicates more independent directions or dimensions.

In terms of matrices, linear independence of rows and columns assures that each dimension adds new information.

This concept is crucial in understanding the form \( \begin{bmatrix} I_k & P \ 0 & Q \end{bmatrix} \) as it highlights that the identity submatrix \( I_k \) represents \( k \) linearly independent columns and rows, implying that the rank of the original matrix is at least \( k \).