Problem 91 - College Algebra

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Find values of \(a\) for which the following matrix is not invertible: $$\left[\begin{array}{rr}1 & a+1 \\\a-2 & 4\end{array}\right]$$

Step-by-Step Solution

Answer

The values of \(a\) for which the given matrix is not invertible, i.e., the roots of the quadratic equation \(-a^2 + a + 2 = 0\), can be calculated using the quadratic formula.

Step 1: Symbolically compute the determinant

Compute the determinant of the matrix by applying the formula \(det(A) = a*d - b*c\). In our case: \(det(A) = 1*4 - (a+1)*(a-2)\). This simplifies to: \(det(A) = 4 - a^2 + a - 2\). So, \(det(A) = -a^2 + a + 2\).

Step 2: Set determinant equal to zero

A matrix is not invertible if its determinant equals zero. Therefore, we need to solve the equation \(-a^2 + a + 2 = 0\). This is a quadratic equation in the form of \(ax^2 + bx + c = 0\).

Step 3: Solve the quadratic equation

We solve the quadratic equation \(-a^2 + a + 2 = 0\) for its roots. The general formula for solving a quadratic equation \(ax^2 + bx + c = 0\) is \(x = [ -b \pm \sqrt{ b^2 - 4ac } ] / (2a)\). By substituting a = -1, b = 1, and c = 2 into this formula, we solve for \(a\).

Step 4: Evaluate the roots

Solutions of the quadratic equation give us the values of \(a\). These are the values for which the determinant of the matrix is zero, hence the matrix isn't invertible for these values of \(a\).

Key Concepts

Determinant

The determinant is a special number that can be calculated from a square matrix. For a 2x2 matrix, the determinant helps us understand important properties of the matrix, like invertibility. In simpler terms, invertibility means if we can "turn the matrix around" to find its reverse or not.
To find the determinant of a 2x2 matrix, follow this formula: If you have a matrix \[\begin{bmatrix}a & b \c & d\end{bmatrix}\]The determinant is calculated by the expression \[ad - bc.\]
This operation will give us a single number that tells us, among other things, whether the matrix is invertible.

Quadratic Equation

A quadratic equation is a type of polynomial equation that takes the form \(ax^2 + bx + c = 0\). Each term represents a part of the equation:

• \(a\) is the coefficient of \(x^2\),
• \(b\) is the coefficient of \(x\),
• \(c\) is a constant term.

The goal is to find the values of \(x\) that make the equation equal to zero.
This equation is called "quadratic" because "quad" means square, referring to the squared term \(x^2\) present in the equation. Solving quadratic equations is important in determining specific conditions, like when a matrix's determinant equals zero to assess invertibility.

Invertibility

A matrix is considered invertible if it has an inverse. The inverse of a matrix is like the reciprocal of a number; when you multiply a matrix by its inverse, you get the identity matrix, similar to how a number times its reciprocal equals 1.
For a 2x2 matrix, the crucial factor in determining invertibility is its determinant.
A square matrix is invertible if and only if its determinant is not zero. Thus, checking if the determinant equals zero directly tells us if reversing the effect of the matrix is possible or not.

Roots of Equations

The roots of an equation are the values of the variable that satisfy the equation, making it equal to zero.
In the context of a quadratic equation like \(ax^2 + bx + c = 0\), the roots are the solutions we find that make this equation true.
To determine these roots, we use the quadratic formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula helps us find the two possible values for \(x\) that will balance the equation to zero.

• If the quantity under the square root, \(b^2 - 4ac\), known as the discriminant, is positive, we get two distinct real roots.
• If it's zero, we get exactly one real root.
• If it's negative, the roots are complex numbers.

Knowing how to find roots is crucial for analyzing scenarios like matrix invertibility, depending on specific variable values.