Question

Compute the determinants of the following matrices. a. \(\left[\begin{array}{rr}2 & -1 \\ 3 & 2\end{array}\right]\) b. \(\left[\begin{array}{rr}6 & 9 \\ 8 & 12\end{array}\right]\) c. \(\left[\begin{array}{ll}a^{2} & a b \\ a b & b^{2}\end{array}\right]\) d. \(\left[\begin{array}{cc}a+1 & a \\ a & a-1\end{array}\right]\) c. \(\left[\begin{array}{rr}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right]\) f. \(\left[\begin{array}{rrr}2 & 0 & -3 \\ 1 & 2 & 5 \\ 0 & 3 & 0\end{array}\right]\) g. \(\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right]\) h. \(\left[\begin{array}{lll}0 & a & 0 \\ b & c & d \\ 0 & e & 0\end{array}\right]\) i. \(\left[\begin{array}{lll}1 & b & c \\ b & c & 1 \\ c & 1 & b\end{array}\right]\) J. \(\left[\begin{array}{lll}0 & a & b \\ a & 0 & c \\ b & c & 0\end{array}\right]\)

Short answer

a: 7, b: 0, c: 0, d: -1, e: 1, f: -39, g: 0, h: -ade, i: 2bc - b^3 - c^2 - 1, j: 2abc.

Step-by-step solution

Calculate Determinant of 2x2 Matrix (a)

For matrix \( \begin{bmatrix} 2 & -1 \ 3 & 2 \end{bmatrix} \), the determinant is calculated using the formula \( ad - bc \), where \( a = 2 \), \( b = -1 \), \( c = 3 \), and \( d = 2 \). Thus, the determinant is \( 2 \times 2 - (-1) \times 3 = 4 + 3 = 7 \).

Calculate Determinant of 2x2 Matrix (b)

For matrix \( \begin{bmatrix} 6 & 9 \ 8 & 12 \end{bmatrix} \), use the same formula: \( ad - bc \). Here \( a = 6 \), \( b = 9 \), \( c = 8 \), \( d = 12 \). So, the determinant is \( 6 \times 12 - 9 \times 8 = 72 - 72 = 0 \).

Calculate Determinant of 2x2 Matrix (c)

For matrix \( \begin{bmatrix} a^2 & ab \ ab & b^2 \end{bmatrix} \), apply the formula \( ad - bc \) where \( a = a^2 \), \( b = ab \), \( c = ab \), \( d = b^2 \). Thus, the determinant is \( a^2 b^2 - (ab)^2 = a^2 b^2 - a^2 b^2 = 0 \).

Calculate Determinant of 2x2 Matrix (d)

For matrix \( \begin{bmatrix} a+1 & a \ a & a-1 \end{bmatrix} \), use the formula \( ad - bc \). Here, \( a = a+1 \), \( b = a \), \( c = a \), \( d = a-1 \). The determinant is \((a+1)(a-1) - a^2 = a^2 - 1 - a^2 = -1 \).

Calculate Determinant of 2x2 Orthogonal Matrix (e)

For matrix \( \begin{bmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{bmatrix} \), the determinant of a 2x2 orthogonal matrix is \( \cos^2 \theta + \sin^2 \theta = 1 \).

Calculate Determinant of 3x3 Matrix (f)

For matrix \( \begin{bmatrix} 2 & 0 & -3 \ 1 & 2 & 5 \ 0 & 3 & 0 \end{bmatrix} \), use the rule of Sarrus or expansion by minors. Choose the first row to expand: \( 2 \times (2 \times 0 - 5 \times 3) + 0 - 3 \times (1 \times 3 - 2 \times 0) = 2 \times (-15) - 3 \times 3 = -30 - 9 = -39 \).

Calculate Determinant of 3x3 Matrix (g)

For matrix \( \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix} \), use the rule of Sarrus or expansion by minors. The determinant follows: \( 1(5 \times 9 - 6 \times 8) - 2(4 \times 9 - 6 \times 7) + 3(4 \times 8 - 5 \times 7) = 0 - 0 + 0 = 0 \).

Calculate Determinant of 3x3 Matrix (h)

For matrix \( \begin{bmatrix} 0 & a & 0 \ b & c & d \ 0 & e & 0 \end{bmatrix} \), the determinant involves: \( 0 \times |\cdots| + a(c \times 0 - d \times e) - 0 \times |\cdots| = -ade \).

Calculate Determinant of 3x3 Matrix (i)

For matrix \( \begin{bmatrix} 1 & b & c \ b & c & 1 \ c & 1 & b \end{bmatrix} \), expand by the first row: \( 1(c \times b - 1 \times 1) - b(b \times b - 1 \times c) + c(b \times 1 - c \times c) = cb - 1 - b^3 + bc + b - c^2 \). Further simplification yields: \( 2bc - b^3 - c^2 - 1 \).

Calculate Determinant of 3x3 Matrix (j)

For matrix \( \begin{bmatrix} 0 & a & b \ a & 0 & c \ b & c & 0 \end{bmatrix} \), use expansion by minors: \( -(a^2 - bc) \). Thus, the determinant is \( - (a^2 + b^2 + c^2) + 2abc \), simplifying to \( 2abc \).

Key concepts

2x2 matrices

A 2x2 matrix is one of the simplest forms of matrices in linear algebra. It consists of two rows and two columns. Calculating the determinant of a 2x2 matrix is straightforward and crucial for understanding matrix operations.

The determinant provides insight into the matrix's properties, such as invertibility. A 2x2 matrix \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]has a determinant calculated as \(ad - bc\).

For example, let's determine the determinant of the matrix \(\begin{bmatrix} 6 & 9 \ 8 & 12 \end{bmatrix}\). Here, \(a = 6\), \(b = 9\), \(c = 8\), and \(d = 12\).

• Apply the formula: \(ad - bc\)
• Calculate: \(6 \times 12 - 9 \times 8 = 72 - 72 = 0\)

Note that a determinant of 0 indicates this matrix is not invertible, meaning there's no inverse matrix.

Calculating determinants of 2x2 matrices becomes intuitive with practice and is an essential step before dealing with larger matrices.

3x3 matrices

3x3 matrices are a step up in complexity from 2x2 matrices and feature three rows and three columns. These matrices are used in more advanced linear algebra calculations.

The determinant of a 3x3 matrix is vital for understanding matrix transformations in three-dimensional space. There are multiple methods to calculate this determinant, such as the rule of Sarrus or expansion by minors.

Consider the 3x3 matrix:\[\begin{bmatrix} 2 & 0 & -3 \ 1 & 2 & 5 \ 0 & 3 & 0 \end{bmatrix}\]Using expansion by minors or Sarrus, expand along the first row:

• Calculate: \(2 \times (2 \times 0 - 5 \times 3) + 0 \times (\cdots) - 3 \times (1 \times 3 - 2 \times 0)\)
• Simplify: \(2 \times (-15) - 3 \times 3 = -30 - 9 = -39\)

This determinant of -39 indicates a unique solution to a related system of linear equations.

Developing the skill of calculating the determinant of 3x3 matrices is essential for tackling real-world problems in physics, engineering, and computer graphics.

linear algebra

Linear algebra is a branch of mathematics focusing on vectors, matrices, and linear transformations. It's fundamental in understanding and solving linear systems, which appear in various scientific computing contexts.

Matrices are a cornerstone in linear algebra, representing different types of linear transformations. When dealing with matrices, the determinant is a vital tool that helps decode the properties of these transformations. For example, whether a matrix is invertible can be determined by its determinant—a non-zero determinant indicates the presence of an inverse.

• In 2D and 3D transformations, the determinant helps ascertain the scaling effect of a transformation matrix.
• The determinant also plays a role in understanding eigenvalues and eigenvectors, critical for stability analysis in differential equations.

Developing a deep understanding of how determinants function within matrices is crucial in mastering linear algebra. This knowledge eases the way when addressing practical problems in science, technology, engineering, and beyond.