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Chapter 3: Problem 53

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

Short Answer

Expert verified
The solution for the given determinant equation are \(x=-1\) and \(x=4\).

Step by step solution

01

Calculate The Determinant

The determinant of this matrix is calculated by the formula, \((x-1)(x-2) - (2*3)\). We will apply this formula to the given matrix.
02

Simplify The Equation

This gives us \(x^2 -3x + 2 - 6\), which simplifies further to \(x^2 - 3x - 4 = 0\) by combining like terms.
03

Factor The Quadratic

The quadratic \(x^2 - 3x - 4 = 0\) can be factored to \((x-4)(x+1) = 0\).
04

Solve For x

Setting each factor equal to 0 gives the solutions \(x=4\) and \(x=-1\). So, the values of \(x\) that satisfy the equation are \(x={-1, 4}\).

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Most popular questions from this chapter

Find (a) the characteristic equation, (b) the eigenvalues, and (c) the corresponding eigenvectors of the matrix. $$\left[\begin{array}{lll} 2 & 0 & 1 \\ 0 & 3 & 4 \\ 0 & 0 & 1 \end{array}\right]$$

Use Cramer's Rule to solve the system of linear equations, if possible. $$\begin{aligned} -0.4 x_{1}+0.8 x_{2} &=1.6 \\ 2 x_{1}-4 x_{2} &=5.0 \end{aligned}$$

Use a graphing utility or computer software program with matrix capabilities to find the eigenvalues of the matrix. Then find the corresponding eigenvectors. $$\left[\begin{array}{rr} 4 & 3 \\ -3 & -2 \end{array}\right]$$

Determine whether the points are collinear. $$(-1,0),(1,1),(3,3)$$

Find an equation of the line passing through the given points. $$(-4,7),(2,4)$$
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