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Chapter 3: Problem 46

Solve for \(x .\) $$\left|\begin{array}{rr} x-6 & 3 \\ -2 & x+1 \end{array}\right|=0$$

Short Answer

Expert verified
The solutions for \(x\) are 6 and -1.

Step by step solution

01

Write down the equation for the determinant

The determinant of a 2x2 matrix: \(\left|\begin{array}{rr} a & b \ c & d \end{array}\right|\) is computed as: \(\text{Determinant} = a*d - b*c\). In this case, \(a = x-6, b = 3, c = -2, d = x+1\). So we get the equation: \((x-6)(x+1) - 3*(-2) = 0\)
02

Simplify the equation

First, we simplify the brackets: \(x^2 - 6x + x - 6 + 6 = 0\). Combining like terms gives: \(x^2 - 5x -6 = 0\).
03

Factor the equation

The factored form of this equation will be: \((x-6)(x+1) = 0\). We can check this by expanding and simplifying the equation to regain the original form.
04

Solve for \(x\)

Setting each factor equal to zero gives the solutions: \(x-6 = 0\) or \(x+1 = 0\). That leads us to two solutions: \(x = 6\) or \(x = -1\).

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Key Concepts

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

2x2 Matrix
When it comes to linear algebra, the 2x2 matrix is one of the fundamental building blocks. It's represented as a square array of numbers with two rows and two columns, like so:
\[\begin{array}{cc} a & b \ c & d \end{array}\]
Each letter represents a number or variable. 2x2 matrices can be used to solve systems of equations, perform transformations in graphics, and various other applications in mathematics and computer science. Understanding how to manipulate and calculate with these matrices is crucial for many advanced topics in these fields.
Determinant of a Matrix
The determinant is a special scalar value that can be computed from the elements of a square matrix. For a 2x2 matrix, it provides information about the matrix such as whether it's invertible or not, and is used in solving systems of linear equations, finding areas of parallelograms, and more.
To calculate the determinant of a 2x2 matrix, you use the simple formula
\[\text{Determinant} = ad - bc\]
where \( a, b, c, d \) are the elements of the matrix. If the determinant is zero, it indicates that the matrix does not have an inverse and that the system might be dependent or inconsistent, depending on the context of the problem.
Factoring Quadratic Equations
Factoring is a method used to solve quadratic equations, which are polynomials of the second degree typically in the form \( ax^2 + bx + c = 0 \). Essentially, factoring transforms the quadratic equation into a product of binomials. Here is a step-by-step approach:
  1. Firstly, identify if the equation is factorable, which is often determined using the discriminant \(b^2 - 4ac\).
  2. Find two numbers that multiply to give \(ac\) and add to give \(b\).
  3. Break the middle term, \(bx\), using the two numbers found and group the terms to factor by grouping.
  4. Lastly, set each factor equal to zero and solve for \(x\) to find the roots of the equation.

An important note is that not every quadratic equation can be factored using integers, and in such cases, you might need to use the quadratic formula or complete the square method to find the solutions.

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Most popular questions from this chapter

Evaluate each determinant when \(a=1, b=4,\) and \(c=-3\). $$\text { (a) }\left|\begin{array}{ccc}0 & b & 0 \\\a & 0 & 0 \\\0 & 0 & c\end{array}\right| \quad \text { (b) }\left|\begin{array}{ccr}a & 0 & 1 \\\0 & c & 0 \\\b & 0 & -16 \end{array}\right|$$

Use a graphing utility to determine whether \(A\) is orthogonal. Then verify that \(|A|=\pm 1\). $$A=\left[\begin{array}{rrr}\frac{3}{5} & 0 & -\frac{4}{5} \\\0 & 1 & 0 \\\\\frac{4}{5} & 0 & \frac{3}{5}\end{array}\right]$$

Guided Proof Prove Property 3 of Theorem 3.3 When \(B\) is obtained from \(A\) by multiplying a row of \(A\) by a nonzero constant \(c, \operatorname{det}(B)=c \operatorname{det}(A)\) Getting Started: To prove that the determinant of \(B\) is cqual to \(c\) times the determinant of \(A,\) you need to show that the determinant of \(B\) is equal to \(c\) times the cofactor expansion of the determinant of \(A\). (i) Begin by letting \(B\) be the matrix obtained by multiplying \(c\) times the \(i\) th row of \(A\) (ii) Find the determinant of \(B\) by expanding in this ith row. (iii) Factor out the common factor \(c .\) (iv) Show that the result is \(c\) times the determinant of \(A\)

Determine whether the points are coplanar. $$(-3,-2,-1),(2,-1,-2),(-3,-1,-2),(3,2,1)$$

Find the volume of the tetrahedron with the given vertices. $$(3,-1,1),(4,-4,4),(1,1,1),(0,0,1)$$
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