Question

Test whether the following matrices are nonsingular: \((a)\left[\begin{array}{rrr}4 & 0 & 1 \\ 19 & 1 & -3 \\ 7 & 1 & 0\end{array}\right]\) (b) \(\left[\begin{array}{rrr}4 & -2 & 1 \\ -5 & 6 & 0 \\ 7 & 0 & 3\end{array}\right]\) (c) \(\left[\begin{array}{rrr}7 & -1 & 0 \\ 1 & 1 & 4 \\ 13 & -3 & -4\end{array}\right]\) \((d)\left[\begin{array}{rrr}-4 & 9 & 5 \\ 3 & 0 & 1 \\ 10 & 8 & 6\end{array}\right]\)

Short answer

All matrices (a), (b), (c), and (d) are nonsingular as their determinants are non-zero.

Step-by-step solution

Calculate the Determinant of Matrix (a)

To determine if a matrix is nonsingular, we compute its determinant. For matrix \(A = \left[\begin{array}{rrr} 4 & 0 & 1 \ 19 & 1 & -3 \ 7 & 1 & 0 \end{array}\right]\), we use the formula for the determinant of a 3x3 matrix:\[\text{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg)\]Substituting the values,\[= 4((1)(0) - (-3)(1)) - 0(19 \cdot 0 - (-3)(7)) + 1(19 \cdot 1 - 1 \cdot 7)\]\[= 4(0 + 3) + 0 + 1(19 - 7)\]\[= 12 + 12 = 24\]Since the determinant of the matrix is not 0, matrix (a) is nonsingular.

Calculate the Determinant of Matrix (b)

For matrix \(B = \left[\begin{array}{rrr} 4 & -2 & 1 \ -5 & 6 & 0 \ 7 & 0 & 3 \end{array}\right]\), compute the determinant:\[\text{det}(B) = 4((6)(3) - (0)(0)) - (-2)((-5)(3) - (0)(7)) + 1((-5)(0) - (6)(7))\]\[= 4(18) + 2(15) - 1(0 - 42)\]\[= 72 + 30 + 42 = 144\]Since the determinant is not 0, matrix (b) is nonsingular.

Calculate the Determinant of Matrix (c)

For matrix \(C = \left[\begin{array}{rrr} 7 & -1 & 0 \ 1 & 1 & 4 \ 13 & -3 & -4 \end{array}\right]\), compute the determinant:\[\text{det}(C) = 7((1)(-4) - (4)(-3)) - (-1)((1)(-4) - (4)(13)) + 0((1)(-3) - (1)(13))\]\[= 7(-4 + 12) + 1(44 - 4) + 0\]\[= 7(8) + 40\]\[= 56 + 40 = 96\]Since the determinant is not 0, matrix (c) is nonsingular.

Calculate the Determinant of Matrix (d)

For matrix \(D = \left[\begin{array}{rrr} -4 & 9 & 5 \ 3 & 0 & 1 \ 10 & 8 & 6 \end{array}\right]\), compute the determinant:\[\text{det}(D) = -4((0)(6) - (1)(8)) - 9((3)(6) - (1)(10)) + 5((3)(8) - (0)(10))\]\[= -4(0 - 8) - 9(18 - 10) + 5(24)\]\[= -4(-8) - 9(8) + 120\]\[= 32 - 72 + 120 = 80\]Since the determinant is not 0, matrix (d) is nonsingular.

Key concepts

Determinant Calculation

When dealing with matrices in mathematics, especially in linear algebra, the determinant plays a crucial role. It is a special number that can be calculated from a square matrix. Determinants help us determine whether a matrix is inversible (nonsingular) or not. For a matrix to be nonsingular, the determinant must not be zero. If it is zero, the matrix is singular, meaning it does not have an inverse.

To calculate the determinant of a 3x3 matrix, the formula is:

• det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)

This formula involves performing operations on the elements of the matrix. Starting from the first row, you multiply the elements by the determinants of the 2x2 matrices that remain after removing the row and column of each element. This process is known as the cofactor expansion.

The calculated determinant is useful for checking properties like invertibility and helping solve systems of linear equations. It is an essential concept in Matrix Algebra and delivers foundational knowledge for more complex operations involving matrices.

Matrix Algebra

Matrix Algebra is the branch of mathematics that deals with matrix operations. Matrices are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns. They serve as a tool for solving problems in various areas, including engineering, physics, and computer science.

In Matrix Algebra, we can perform various operations including addition, subtraction, and multiplication of matrices. However, not all operations work the same way as they do with regular numbers. For instance, matrix multiplication is not commutative, meaning \(AB eq BA\) in general.

• Addition: Matrices can be added or subtracted if they share the same dimensions.
• Multiplication: Matrices can be multiplied if the number of columns in the first matrix matches the number of rows in the second.

Another important operation is finding the inverse of a matrix, which is only possible if the matrix is nonsingular. The inverse of a matrix, when multiplied by the original matrix, results in the identity matrix.

Understanding matrix algebra is crucial because it provides the framework for various mathematical models and helps us deal with linear transformations and systems of linear equations.

3x3 Matrix

A 3x3 matrix is a specific type of square matrix composed of three rows and three columns. These matrices are prevalent in studies involving linear transformations and provide a compact way to express systems of linear equations. Each element in a 3x3 matrix has a specific influence in the linear transformation it represents.

A simplified view of a 3x3 matrix looks like this:

• \[\begin{bmatrix}a_{11} & a_{12} & a_{13} \a_{21} & a_{22} & a_{23} \a_{31} & a_{32} & a_{33}\end{bmatrix}\]

Each "a" with its subscript is an element of the matrix, where the first subscript indicates the row and the second indicates the column.

The determinant and the properties derived from it, such as whether or not the matrix has an inverse, are vital. Calculating the determinant of a 3x3 matrix involves using a standard formula that utilizes its nine elements and helps confirm whether the system represented by the matrix can be resolved uniquely. Once we know it's nonsingular, the matrix can be used in further matrix operations and applications, such as solving equations or applying geometric transformations.