91Ó°ÊÓ

A 2x2 matrix is the simplest form of a square matrix. It has two rows and two columns. Matrices can represent systems of equations, transformations in geometry, and more.

In this exercise, we have a 2x2 matrix given by:
\begin{bmatrix} x & 4 \ x & x^2 \ \text In this matrix, each entry has a specific position indicated by its row and column. For instance, \(x\) is in the first row and first column, while \(4\) is in the first row and second column.

Understanding how to work with 2x2 matrices is a fundamental skill in linear algebra, and it's especially important when learning how to find determinants.

The determinant of a matrix can tell us several things, such as whether the matrix is invertible, and it plays a key role in solving systems of linear equations. For a 2x2 matrix:
\begin{bmatrix} a & b \ c & d \ \text the determinant is calculated using the formula: \text \text {det = ad - bc.}

Here, \(a\), \(b\), \(c\), and \(d\) represent the elements of the matrix. To find the determinant, you multiply the elements of the main diagonal (\(a\) and \(d\)) and subtract the product of the other diagonal's elements (\(b\) and \(c\)).

Applying this to our matrix:\begin{bmatrix} x & 4 \ x & x^2 \ \text you would identify \(a = x\), \(b = 4\), \(c = x\), and \(d = x^2\). Plugging these into the formula, we get:
\text {det = x \times x^2 - 4 \times x.}

Simplification is a crucial step in finding the determinant.

Once we substitute the values into the determinant formula, we get: \text \text {det = x \times x^2 - 4 \times x}

Simplifying this expression involves applying basic algebraic rules, such as distributing and combining like terms. First, we calculate \(x \times x^2\) to obtain \(x^3\). Then we calculate \(4 \times x\) to get \(4x\):
\text \text {det = x^3 - 4x.}

This expression \text represents the determinant of our given matrix after simplification: \text \text {det = x^3 - 4x.}

Remember, understanding how to simplify expressions is essential in various areas of mathematics, including working with determinants.