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Chapter 4: Problem 36

Find the determinants assuming that \\[\left|\begin{array}{lll}a & b & c \\\d & e & f \\\g & h & i \end{array}\right|=4.\\] \(\left|\begin{array}{ccc}3 a & -b & 2 c \\ 3 d & -c & 2 f \\ 3 g & -h & 2 i\end{array}\right|\)

Short Answer

Expert verified
The determinant is 24.

Step by step solution

01

Identify the Matrix

Identify the given matrix and recognize its format. We have a 3x3 matrix with elements \( \left| \begin{array}{ccc} 3a & -b & 2c \ 3d & -c & 2f \ 3g & -h & 2i \end{array} \right| \).
02

Use Scalar Multiplication Property

Recall that when each element of a row or column of a determinant is multiplied by a scalar, the determinant is multiplied by that scalar as well. Here, notice that the first column of the determinant is \(3(a, d, g)^T\). Thus, the determinant can be expressed as being multiplied by scalar 3 for the entire first column.
03

Factor Out Scalings From Columns

Similarly, notice that each element in the third column is multiplied by 2. Thus, factor out 2 from the elements in the third column.
04

Calculate the Determinant

The determinant becomes the product of the scalars and the determinant of the original matrix. Therefore, \( 3 \times 2 \times \left| \begin{array}{ccc} a & b & c \ d & e & f \ g & h & i \end{array} \right| \).
05

Substitute and Solve

Substitute the given determinant value \( \left| \begin{array}{ccc} a & b & c \ d & e & f \ g & h & i \end{array} \right| = 4 \). Therefore, the new determinant is evaluated as \( 3 \times 2 \times 4 = 24 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

3x3 matrices
In linear algebra, matrices are essential for organizing numbers in a structured and systematic way. When we specifically talk about 3x3 matrices, these matrices consist of three rows and three columns, forming a square matrix. Each element in a matrix is often denoted by a letter, such as 'a', 'b', or 'c'. These elements are arranged inside brackets, highlighting the organized nature of data in a 3x3 matrix.
For the exercise at hand, consider the matrix:\[\begin{array}{ccc}3a & -b & 2c \3d & -c & 2f \3g & -h & 2i \end{array}\]Here, each element represents a number or expression placed at the intersection of rows and columns. We begin analyzing this matrix to calculate its determinant, which is a special number associated with the matrix.
scalar multiplication property
Scalar multiplication is a straightforward yet powerful property used in determinant calculations. It relates to the multiplication of each element of a row or column by a constant value, known as a scalar. This changes the determinant of the whole matrix by the same scalar factor.
For instance, if you multiply every number in a row by 3, the entire determinant gets multiplied by 3. In the given exercise, the scalar multiplication happens:
  • In the first column: every element has been multiplied by 3.
  • In the third column: every element multiplied by 2.
These scalars can be factored out of their respective columns. The determinant of the 3x3 matrix thus becomes multiplied by both scalars. After factoring out, these scalars are then multiplied with the original determinant of the matrix.
determinant calculation
To find the determinant of a matrix is to locate the specific scalar value that provides critical insights into the matrix's properties, such as solvability of linear equations. The determinant of a 3x3 matrix is calculated using a specific formula in more general scenarios, yet for the current exercise, simplification through scalar factors has been employed.
The calculation can be broken into steps:
  • Start with recognizing the scalar multiplication of certain columns by 3 and 2.
  • Express the determinant as a product of these scalars with the original determinant.
For the matrix given,\[3 \times 2 \times \left|\begin{array}{ccc}a & b & c \d & e & f \g & h & i\end{array}\right| = 3 \times 2 \times 4\] equals 24. Hence, breaking down the effect of scalar multiplication is crucial in simplifying determinant calculation.

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Most popular questions from this chapter

Find an invertible matrix Pand a matrix \(C\) of the form \(C=\left[\begin{array}{cr}a & -b \\ b & a\end{array}\right]\) such that \(A=P C P^{-1}\) Sketch the first six points of the trajectory for the dynamical system \(\mathbf{x}_{\mathrm{k}+1}=A \mathbf{x}_{\mathrm{k}}\) with \(\mathbf{x}_{0}=\left[\begin{array}{l}1 \\ 1\end{array}\right]\) and classify the origin as \(a\) spiral attractor, spiral repeller, or orbital center $$A=\left[\begin{array}{ll} 0 & -1 \\ 1 & \sqrt{3} \end{array}\right]$$

It can be shown that a nonnegative \(n \times n\) matrix is irreducible if and only if \((I+A)^{n-1}>0 .\) Use this criterion to determine whether the matrix \(A\) is irreducible. If \(A\) is reducible, find a permutation of its rows and columns that puts \(A\) into the block form \\[ \left[\begin{array}{ll} B & C \\ O & D \end{array}\right] \\] $$A=\left[\begin{array}{lllll} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 \end{array}\right]$$

Write out the first six terms of the sequence defined by the recurrence relation with the given initial conditions. $$y_{0}=0, y_{1}=1, y_{n}=y_{n-1}-y_{n-2} \text { for } n \geq 2$$

Consider the dynamical system \(\mathbf{x}_{k+1}=A \mathbf{x}_{k}\) (a) Compute and plot \(\mathbf{x}_{0}, \mathbf{x}_{1}, \mathbf{x}_{2}, \mathbf{x}_{3}\) for \(\mathbf{x}_{0}=\left[\begin{array}{l}1 \\\ 1\end{array}\right]\) (b) Compute and plot \(\mathbf{x}_{0}, \mathbf{x}_{1}, \mathbf{x}_{2}, \mathbf{x}_{3}\) for \(\mathbf{x}_{0}=\left[\begin{array}{l}1 \\\ 0\end{array}\right]\) (c) Using eigenvalues and eigenvectors, classify the origin as an attractor, repeller, saddle point, or none of these. (d) Sketch several typical trajectories of the system. $$A=\left[\begin{array}{rr} 2 & -1 \\ -1 & 2 \end{array}\right]$$

In Exercises \(59-64\), find the general solution to the given system of differential equations. Then find the specific solution that satisfies the initial conditions. (Consider all functions to be functions of \(t .\) $$\begin{array}{ll} x^{\prime}=x+3 y, & x(0)=0 \\ y^{\prime}=2 x+2 y, & y(0)=5 \end{array}$$
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