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Chapter 4: Problem 20

Suppose that \(M \in M_{n \times n}(F)\) can be written in the form $$ M=\left(\begin{array}{ll} A & B \\ O & I \end{array}\right) $$ where \(A\) is a square matrix. Prove that \(\operatorname{det}(M)=\operatorname{det}(A)\).

Short Answer

Expert verified
In order to prove $\operatorname{det}(M) = \operatorname{det}(A)$, we first use the Laplace expansion on the determinant of M and then apply the properties of determinants. After expanding, since all entries in the block matrix O are zero, their product with the determinant of any cofactor matrix is also zero. Therefore, the determinant of M reduces to the determinant of its submatrix A, and we can conclude that $\operatorname{det}(M) = \operatorname{det}(A)$.

Step by step solution

01

Recall the definition of a block matrix.

A block matrix is a matrix that is partitioned into smaller matrices, called blocks, in such a way that the entries of the original matrix are preserved. In this exercise, we are given the block matrix M: $$ M = \begin{pmatrix} A & B \\ O & I \end{pmatrix} $$ where A is a square matrix, B and O are matrix blocks, and I is the identity matrix.
02

Understanding the determinant and its properties.

The determinant is a scalar value that relates to a square matrix. For a 2x2 matrix A: $$ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} $$ its determinant would be given by: $$ \operatorname{det}(A) = ad - bc $$ Some important properties of determinants are: 1. The determinant is linear in each row and each column. This means that for a given row or column, the determinant can be written as the sum of several determinants, where each one has one term from the given row or column. 2. If two rows or columns of a matrix are equal, the determinant is zero. 3. If a matrix has a row or column consisting only of zeros, the determinant is zero. We will use these properties to prove our statement.
03

Calculate the determinant of M using Laplace expansion.

Laplace expansion is a technique used to calculate the determinant of a matrix. We will apply it to our matrix M. Let's expand the determinant along its last row, which consists of matrix block O and the identity matrix I: $$ \operatorname{det}(M) = \sum_{i=1}^{n} (-1)^{i+n}[\text{O}_{in}]\operatorname{det}(M_{i,n}) $$ where \(\text{O}_{in}\) is the entry of the matrix O in the \(i-th\) row and n-th column and \(M_{i,n}\) is the cofactor matrix obtained after eliminating the \(i-th\) row and n-th column of M.
04

Demonstrate that the determinant of M equals the determinant of A using the properties of determinants.

We will now prove that \(\operatorname{det}(M) = \operatorname{det}(A)\). According to the Laplace expansion we found in step 3, we need to calculate the values of \([\text{O}_{in}]\operatorname{det}(M_{i,n})\). Note that since every entry \([\text{O}_{in}]\) is a zero entry, their product with the determinant of any cofactor matrix is always zero. Therefore, we can conclude that the determinant of M can be reduced to the determinant of its submatrix A, giving us the result: $$ \operatorname{det}(M) = \operatorname{det}(A) $$ We have proven the statement.

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Most popular questions from this chapter

Determine whether the following vectors in \(\mathbf{R}^{4}\) are linearly dependent or independent: (a) \(\quad(1,2,-3,1),(3,7,1,-2),(1,3,7,-4)\)' (b) \(\quad(1,3,1,-2),(2,5,-1,3),(1,3,7,-2)\).

Find the dimension and a basis of the subspace \(W\) of \(\mathbf{M}=\mathbf{M}_{2,3}\) spanned by \\[A=\left[\begin{array}{lll} 1 & 2 & 1 \\ 3 & 1 & 2 \end{array}\right], \quad B=\left[\begin{array}{lll} 2 & 4 & 3 \\ 7 & 5 & 6 \end{array}\right], \quad C=\left[\begin{array}{lll} 1 & 2 & 3 \\ 5 & 7 & 6 \end{array}\right].\\]

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