Problem 3 - Precalculus

Question

Find the inverse of the matrix or state that the matrix is not invertible. $$ C=\left[\begin{array}{rr} 6 & 15 \\ 14 & 35 \end{array}\right] $$

Step-by-Step Solution

Answer

The matrix \( C \) is not invertible because its determinant is 0.

Step 1: Check if the matrix is square

The given matrix \( C \) is a 2x2 matrix: \( \begin{bmatrix} 6 & 15 \ 14 & 35 \end{bmatrix} \). It has two rows and two columns, so it is a square matrix.

Step 2: Calculate the determinant

The determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated as \( ad - bc \). For our matrix \( C \), the determinant is \( 6 \times 35 - 15 \times 14 \). Calculate each product: \( 6 \times 35 = 210 \) and \( 15 \times 14 = 210 \). Thus, the determinant is \( 210 - 210 = 0 \).

Step 3: Determine invertibility based on determinant

A square matrix is invertible if and only if its determinant is non-zero. Since the determinant of matrix \( C \) is 0, matrix \( C \) is not invertible.

Key Concepts

Square Matrix

A square matrix is a simple yet crucial concept in linear algebra. It is defined as a matrix with an equal number of rows and columns.
This property is essential because it forms the basis for many matrix operations, including finding an inverse. **Square matrices** can be of size 2x2, 3x3, or more, depending on the number of rows and columns.

• For example, a **2x2 matrix** has two rows and two columns.
• A **3x3 matrix** has three rows and three columns, and so forth.

The importance of square matrices arises in determining invertibility, as only square matrices can have inverses. For the matrix to potentially have an inverse, being square is a necessary initial check.

Determinant

The determinant is a special number calculated from a square matrix. It offers insight into several properties of the matrix, including invertibility. For a 2x2 matrix, the determinant is calculated using the formula:

• \[Determinant = ad - bc\]
• Where \( a \), \( b \), \( c \), and \( d \) are the elements of the matrix arranged as:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]

The determinant has a geometric interpretation, giving a sense of the volume scaling factor when the matrix is viewed as a transformation matrix.
If the determinant is zero, it indicates that the matrix squashes the space it transforms into a lower dimension, thus making it non-invertible.

Invertibility

Invertibility is a property that tells us whether a square matrix has an inverse. The inverse of a matrix is like a reverse button: applying the inverse will undo the effect of the original matrix. However, not all matrices have inverses.
The key factor in determining whether a matrix is invertible is its determinant.

• A matrix is **invertible (or non-singular)** if and only if its determinant is non-zero.
• If the determinant is zero, the matrix is **non-invertible (or singular)**.

This criterion explains why checking the determinant is a standard step in these calculations. If the determinant equals zero, as seen in our example matrix, the absence of an inverse implies certain transformations cannot be undone or reverted.

2x2 Matrix

2x2 matrices are the simplest form of square matrices, yet hold significant importance in linear algebra.
They consist of four elements arranged in two rows and two columns, which makes them easy to compute by hand compared to larger matrices.

• A typical 2x2 matrix looks like:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]

These matrices are widely used in various applications, from solving systems of linear equations to performing rotations and scaling in graphics.
Finding the **determinant** is straightforward, and for lattice transformations, the determinant gives an essential geometric interpretation, often relating to area or scaling factors. If the determinant is non-zero, you can compute the inverse using:\[ \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \times \frac{1}{ad-bc}\]