Problem 91 - Algebra and Trigonometry

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: $$ \left[\begin{array}{cc} {1} & {a+1} \\ {a-2} & {4} \end{array}\right] $$

Step-by-Step Solution

Answer

The values of \(a\) for which the matrix is not invertible are \(a = 0\) and \(a = 2\).

Step 1: Calculate the Determinant

The determinant of a 2x2 matrix \( \left[\begin{array}{cc}a & b \\c & d \end{array}\right]\) can be found using the rule \(ad - bc\). Use this rule and calculate the determinant for the given matrix. This results in the determinant \(D = (1*4) - ((a+1)*(a-2)) = 4 - a^2 + 2a + 3A - 4\).

Step 2: Set the Determinant equal to Zero

To find the values of \(a\) for which the matrix is not invertible, the determinant of the matrix must be equal to zero. In other words, the equation \(D = 0\) must be solved for \(a\). This implies \(a^2 - 2a = 0\).

Step 3: Solve the Equations

The equation \(a^2 - 2a = 0\) can be factored into \(a*(a - 2) = 0\). The solutions to this equation are \(a = 0\) or \(a = 2\). These are the values of \(a\) for which the matrix is not invertible.

Key Concepts

Determinant of a Matrix

The determinant of a matrix is like a mathematical 'passport' that allows a matrix to travel to the world of invertibility. It's a special number that we can calculate from the elements of a square matrix. To understand why it's important, think of the determinant as a gatekeeper; when it's not zero, it lets a matrix become invertible, which means we can find a unique matrix that acts like its 'undo button'.

The simplest case to visualize is a 2x2 matrix, like the one in our exercise. For such a matrix, the determinant is found using a straightforward formula: \(ad-bc\), where 'a', 'b', 'c', and 'd' are the elements of the matrix. If we plug in the elements from our exercise's matrix, we get \(1 \times 4 - (a+1)(a-2)\). This calculation gives us a quadratic expression in terms of 'a', and if this expression equals zero, our matrix's 'passport' gets revoked—it cannot be inverted.

Matrix Algebra

Matrix algebra is a handy toolkit that enables us to perform operations like addition, subtraction, multiplication, and finding inverses of matrices. Just like with numbers, these operations help us solve a variety of problems, from simple puzzles to complex science and engineering challenges.

While numbers have straightforward rules, matrix operations follow more elaborate patterns. For instance, when we multiply matrices, we don't do it element by element; we calculate the sum of products of rows and columns. But why bother with all these peculiar methods? Because matrices are powerful in representing systems, transformations, and much more in a compact form. By mastering matrix algebra, we unlock the ability to handle linear transformations and solve systems of linear equations effortlessly.

Solving Linear Equations

Whenever we come across a linear equation, it's like facing a lock. Solving the equation is akin to finding the right key to open it up and reveal the treasures hidden inside—our solutions. In the case of our matrix exercise, the 'treasure' is knowing when the matrix can't be inverted.

By setting the determinant to zero, we crafted a linear equation, \(a^2 - 2a = 0\), which we needed to solve to find the values of 'a' that slam shut the gate to invertibility. Factoring the quadratic expression gave us two 'keys': \(a = 0\) and \(a = 2\). Plugging in these values into our matrix 'lock' means that for these specific 'a's, our matrix loses its power to be inverted—it simply can't budge the lock open. Understanding how to solve such equations is crucial, as it not only helps in algebra but also prepares the ground for more advanced mathematical endeavors.