/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 240 Define submatrix and subdetermin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 240

Define submatrix and subdeterminant.

Short Answer

Expert verified
A submatrix is formed by deleting any row and column from a given matrix, maintaining the relative position of the remaining elements. A subdeterminant is the determinant of a submatrix. To calculate the subdeterminant, find the submatrix by removing a specific row and column, then compute the determinant of the submatrix. For example, with matrix \(A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}\), the submatrix \(A_{22} =\begin{pmatrix} 1 & 3 \\ 7 & 9 \end{pmatrix}\), and the subdeterminant is \(\text{det}(A_{22}) = -12\).

Step by step solution

01

Define a matrix

A matrix is a rectangular array of numbers arranged in rows and columns. The entries in the matrix are called elements, and the size of a matrix is described by identifying the number of rows and columns. For example, a 3x3 matrix has 3 rows and 3 columns.
02

Define a determinant

The determinant is a scalar value that can be computed only from a square matrix (i.e., the number of rows is equal to the number of columns). It is a special number that helps in solving systems of linear equations and finding the inverse of a matrix. The determinant of a 2x2 matrix \(A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\) is given by \(\text{det}(A) = ad - bc\).
03

Define a submatrix

A submatrix is formed by deleting any row and column from a given matrix. For instance, given a matrix \(A\), you can create a submatrix by removing the \(i^{\text{th}}\) row and the \(j^{\text{th}}\) column. In other words, a submatrix is a matrix that consists of a subset of the elements of the original matrix but still maintains the relative position of those elements.
04

Define a subdeterminant

A subdeterminant is the determinant of a submatrix. To calculate the subdeterminant, first, find the submatrix by removing a specific row and column from the original matrix. Then, compute the determinant of the submatrix.
05

Example of submatrix and subdeterminant calculation

Let's consider a 3x3 matrix \(A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}\). To find the submatrix \(A_{22}\) (submatrix formed by deleting the 2nd row and 2nd column), we get: \(A_{22} =\begin{pmatrix} 1 & 3 \\ 7 & 9 \end{pmatrix}\) Now, we can find the subdeterminant, which is the determinant of the submatrix. \(\text{det}(A_{22}) = (1)(9) - (3)(7) = 9 - 21 = -12\) Thus, the subdeterminant of matrix \(A\) when removing the 2nd row and the 2nd column is -12.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

\(\begin{aligned}&\text { Find } A-B \text { if } \\\&\qquad A=\left|\begin{array}{rrr}3 & -2 & 5 \mid \text { and } B=\mid 2 & 3 & 2 \mid \\\ \mid-1 & 2 & 3\end{array}\right| & \mid-3 & 4 & 6\end{aligned} \mid\).

Find \(\mathrm{f}(\mathrm{A})\) where \(\mathrm{A}=|1 \underset{\mid 4}{\mid 1}-2|\) and \(\mathrm{f}(\mathrm{t})=\mathrm{t}^{2}-3 \mathrm{t}+7\)

What is the angle between a diagonal of a cube and one of its edges?

Given $$ \mathrm{A}=\mid \begin{array}{cc} 1 & \mathrm{e}^{\mathrm{t}} \mid \\ \mid \mathrm{t}^{2} & \mathrm{t} \end{array} $$ find \({ }^{1} \int_{0} \mathrm{~A}(\mathrm{t}) \mathrm{d} \mathrm{t}\)

Find the inverse of A where $$ \mathrm{A}=\mid \begin{array}{cc} 2 & 3 \\ \mid 3 & 5 \mid \end{array} $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.