Problem 2 - College Algebra with Modeling and Visualization

Question

Determine if the matrix \(A\) is invertible by cal. culating det \(A\) $$ A=\left[\begin{array}{rr} 1 & -3 \\ 2 & 6 \end{array}\right] $$

Step-by-Step Solution

Answer

The matrix is invertible because its determinant is 12, which is not zero.

Step 1: Identify the formula for the determinant of a 2x2 matrix

For a matrix \(A\) of the form \(\left[\begin{array}{cc} a & b \ c & d \end{array}\right]\), the determinant \(\det(A)\) is calculated using the formula \(ad - bc\).

Step 2: Assign values from matrix A to variables

From the matrix \(A=\left[\begin{array}{rr} 1 & -3 \ 2 & 6 \end{array}\right]\), assign \(a=1\), \(b=-3\), \(c=2\), and \(d=6\).

Step 3: Calculate the determinant using the formula

Substitute the values into the formula: \(\det(A) = (1 \times 6) - (-3 \times 2)\). Calculate inside the brackets first, resulting in \(6 + 6 = 12\).

Step 4: Determine if the determinant is non-zero

Since \(\det(A) = 12\), which is not equal to zero, the matrix \(A\) is invertible.

Key Concepts

Invertible Matrix

An invertible matrix, also known as a non-singular or non-degenerate matrix, is a square matrix that has an inverse. This means if a matrix is invertible, there exists another matrix which when multiplied with the original matrix, results in the identity matrix. The identity matrix is essentially the equivalent of the number 1 in regular multiplication for matrices.
To simplify, think of an invertible matrix as a lock and its inverse matrix as the key. Only the correct key (inverse) will open or unlock the matrix (resulting in the identity matrix).
For a matrix to be invertible, its determinant must be non-zero. This is a crucial test to check before attempting any calculations involving the inverse matrix.

2x2 Matrix Determinant

To determine if a 2x2 matrix is invertible, you need to calculate its determinant. The determinant serves as a special number that can tell you a lot about the matrix.
For a 2x2 matrix, the determinant is found using a simple formula:

• For a matrix \[\begin{bmatrix} a & b \ c & d \end{bmatrix} \], the determinant \( ext{det}(A)\) is calculated using: \(ad - bc\).

Let’s say you have a matrix:\[A = \begin{bmatrix} 1 & -3 \2 & 6 \end{bmatrix} \]
Here, you assign the elements of the matrix to variables "a", "b", "c", and "d". Then, simply plug these numbers into the formula:
\( ext{det}(A) = 1 \times 6 - (-3 \times 2)\)
This calculation results in \(12\), indicating the matrix is indeed invertible as the determinant is not zero.

Algebra

In algebra, matrices are utilized for numerous applications in solving systems of equations, transformations, and more. Understanding the fundamental concept of determinants is crucial.
Calculating determinants involves basic arithmetic operations. It requires one to be attentive when dealing with matrix elements. This might involve handling negative numbers or multiplication operations which could affect your final result.
In a broader sense, algebra allows you to manipulate mathematical expressions to isolate variables, find solutions to equations, or even prove certain properties of mathematical systems. The determinant calculation is just one aspect, but a foundational one, especially in linear algebra where matrix operations take precedence. Matrices are powerful tools, and algebra provides the language and framework to unlock their potential.