Question

Find the determinant of the \(2 \times 2\) matrix. \(\left[\begin{array}{cc}10 & -10 \\ -10 & 0\end{array}\right]\)

Short answer

The determinant is -100.

Step-by-step solution

Identify the Matrix

We are given a 2x2 matrix: \[ A = \begin{bmatrix} 10 & -10 \ -10 & 0 \end{bmatrix} \] Our task is to find the determinant of this matrix.

Apply Determinant Formula for 2x2 Matrix

The determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated using the formula \( ad - bc \). For our matrix \( A \), \( a = 10 \), \( b = -10 \), \( c = -10 \), and \( d = 0 \).

Calculate the Products

Apply the values to the formula: \[ ad = (10) \times (0) = 0 \] \[ bc = (-10) \times (-10) = 100 \]

Compute the Determinant

Substitute the products into the determinant formula: \[ \text{Det}(A) = ad - bc = 0 - 100 = -100 \] The determinant of the matrix is \(-100\).

Key concepts

2x2 Matrix

A 2x2 matrix is a simple but fundamental concept in linear algebra that consists of two rows and two columns. This is the smallest square matrix that can be created. Each element of a 2x2 matrix is a number that fills one of these four positions. For instance, given the matrix:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]- The elements at the first row are \(a\) and \(b\).- The elements at the second row are \(c\) and \(d\).In the context of linear transformations, a 2x2 matrix can represent transformations on a 2-dimensional plane, such as rotation, scaling, or shearing. Understanding this type of matrix is key to studying more complex matrices and mathematical concepts. In problems involving 2x2 matrices, it's crucial to correctly identify each element and its position, as these are necessary for calculations like finding the determinant.

Determinant Calculation

Calculating the determinant is a way to understand some properties of a matrix. For a 2x2 matrix, the determinant gives an insight into whether the matrix is invertible. Calculating the determinant of a 2x2 matrix is straightforward and involves a simple formula. If your matrix is:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]The formula for the determinant of a 2x2 matrix is:\[ ad - bc \]This means you multiply \(a\) by \(d\) and \(b\) by \(c\), then subtract the second product from the first:

• Calculate \(ad\)
• Calculate \(bc\)
• Subtract \(bc\) from \(ad\)

This result tells us important things about the matrix like if it can reverse transformations, indicated by whether its determinant is zero.

Matrix Algebra

Matrix algebra involves operations and rules that include addition, subtraction, and multiplication of matrices. Particularly, understanding the determinant is critical because it touches upon the properties that these operations offer.

• **Addition and Subtraction:** These operations are performed element-wise, meaning elements in corresponding positions are added or subtracted.
• **Multiplication:** Not as straightforward as addition or subtraction, matrix multiplication combines the rows of the first matrix with the columns of the second matrix to produce the answer.
• **Identity Matrices:** These are special matrices which, when multiplied by another matrix, yield the same matrix. For 2x2 matrices, the identity looks like \(\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\).

The foresight offered by the determinant, particularly whether a matrix is invertible, significantly impacts matrix multiplication outcomes and potential solutions to systems of equations. Thus, it forms a cornerstone in the broader field of matrix algebra, which is heavily utilized in physics, engineering, computer graphics, and statistics.