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Chapter 4: Problem 4

In Exercises 1-10, find the determinant of the given matrix. $$ \left[\begin{array}{rrr} 4 & -1 & 2 \\ 3 & 1 & 0 \\ -1 & 2 & 1 \end{array}\right] $$

Short Answer

Expert verified
The determinant of the matrix is 21.

Step by step solution

01

Identify the Matrix

The given matrix is a 3x3 matrix:\[\begin{bmatrix} 4 & -1 & 2 \ 3 & 1 & 0 \ -1 & 2 & 1 \end{bmatrix}\]
02

Apply the Determinant Formula for 3x3 Matrix

The formula for the determinant of a 3x3 matrix \( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \) is:\[\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]
03

Substitute the Values into the Formula

Identify and substitute the corresponding values into the formula:- \( a = 4 \), \( b = -1 \), \( c = 2 \)- \( d = 3 \), \( e = 1 \), \( f = 0 \)- \( g = -1 \), \( h = 2 \), \( i = 1 \)Substitute into the determinant formula:\[4(1 \cdot 1 - 0 \cdot 2) - (-1)(3 \cdot 1 - 0 \cdot -1) + 2(3 \cdot 2 - 1 \cdot -1)\]
04

Calculate the Determinant

Evaluate each term:- First term: \(4(1 - 0) = 4 \)- Second term: \((-1)(3 - 0) = -3 \)- Third term: \(2(6 + 1) = 14 \)Combine the results: \( 4 - (-3) + 14 = 4 + 3 + 14 = 21 \)

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Key Concepts

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

3x3 Matrix
Understanding a 3x3 matrix is crucial in exploring linear algebra since it represents a system with three equations and three variables. The matrix provided in the exercise is a 3x3 matrix that looks like this: \[\begin{bmatrix}4 & -1 & 2 \3 & 1 & 0 \-1 & 2 & 1 \end{bmatrix}\]In simple terms, the matrix has three rows and three columns:
  • The first row is \( [4, -1, 2] \).
  • The second row is \( [3, 1, 0] \).
  • The third row is \( [-1, 2, 1] \).
This layout of a matrix is fundamental in math as it helps organize the numbers, which represent different coefficients connected to variables in equations. Working with matrices like the 3x3 matrix allows you to perform operations that are very useful in solving linear algebra problems.
Determinant Formula
The determinant of a matrix is a special number that can tell you a lot about the matrix itself, like whether it's invertible. The formula for the determinant of a 3x3 matrix with elements labeled as \(a, b, c, d, e, f, g, h, i\) is:\[\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]Using this formula means working through each component of the matrix to evaluate how they affect the overall determinant. This formula stems from expanding the matrix along its first row.
  • The product \( ei - fh \) helps understand the influence of columns and rows by creating smaller sub-matrices.
  • Continue by doing similarly for the other sets of minor determinants: \( di - fg \) and \( dh - eg \).
The determinant calculation is used frequently as it plays a key role in more complex matrix operations, such as finding inverses or solving systems of equations.
Matrix Calculation
After understanding the formula, you plug in the matrix's values to compute its determinant. Let鈥檚 substitute the given matrix values into our formula:
  • First, for the main diagonal \( ei - fh \): calculate \( 1 \times 1 - 0 \times 2 \) which results in \(1\).
  • Next, for \( di - fg \): calculate \( 3 \times 1 - 0 \times -1 \) giving \(3\).
  • Finally, for \( dh - eg \): calculate \( 3 \times 2 - 1 \times -1 \) resulting in \(7\).
Now apply these results within the determinant formula: \[4 \times 1 - (-1) \times 3 + 2 \times 7\]This simplifies to:
  • \( 4 \times 1 = 4 \)
  • \(+ 1 \times 3 = 3\)
  • \(+ 2 \times 7 = 14\)
The result is then \(4 + 3 + 14 = 21\). Thus, the determinant of the matrix is 21.
Linear Algebra Problem Solving
In linear algebra, understanding the principles behind matrix operations is vital for problem-solving. Determinants play a key role in this. When you find the determinant of a 3x3 matrix like the one given, it helps to determine:
  • If the matrix is invertible (if the determinant is non-zero, the matrix is invertible).
  • The solutions of systems of equations that are represented by the matrix.
  • Various transformations in applied mathematics (such as scaling or rotating vectors).
By computing determinants, you can tackle complicated linear equations and understand how linear systems behave. Practicing these calculations strengthens your ability to manipulate and solve linear algebra systems, contributing to your overall problem-solving skills in mathematics.

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Most popular questions from this chapter

Find the determinant of the given matrix. $$ \left[\begin{array}{rrrrr} 0 & 0 & 0 & 3 & -4 \\ 0 & 0 & 0 & 2 & 1 \\ -1 & 2 & 4 & 0 & 0 \\ 3 & 1 & -2 & 0 & 0 \\ 5 & 1 & 5 & 0 & 0 \end{array}\right] $$

In Exercises 41-44, find the volume of the ietrahedron having the given vertices. (Consider how the volume of a tetrahedron having three vectors from one point as edges is related to the volume of the box having the same three vectors as adjacent edges.J \((-3,0,1),(4,2,1),(0,1,7),(1,1,1)\)

in Exercises 1-4, find the indicated determinaut. \(\left|\begin{array}{rr}-1 & 3 \\ 5 & 0\end{array}\right|\)

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Show by direct computation that $$ \left|\begin{array}{lll} a_{1} & a_{2} & a_{2} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{array}\right|=-\left|\begin{array}{lll} a_{1} & a_{2} & a_{3} \\ c_{1} & c_{2} & c_{3} \\ b_{1} & b_{2} & b_{3} \end{array}\right| . $$
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