Problem 38 - Intermediate Algebra

Question

Evaluate each determinant. $$ \left|\begin{array}{lll} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 1 & 1 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant of the matrix is 0.

Step 1: Understand the Determinant Layout

This is a 3x3 matrix, and the determinant of a 3x3 matrix can be found using the formula:\[det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)\]where the matrix is:\[\left| \begin{array}{ccc} a & b & c \ d & e & f \ g & h & i \end{array} \right|\]Here, the matrix given is:\[\begin{array}{ccc}1 & 0 & 1 \0 & 1 & 0 \1 & 1 & 1 \ \end{array}\]

Step 2: Substitute Values into the Formula

From the given matrix, we substitute the values:\( a = 1, b = 0, c = 1, d = 0, e = 1, f = 0, g = 1, h = 1, i = 1 \)into the determinant formula. This becomes:\[det(A) = 1(1*1 - 0*1) - 0(0*1 - 0*1) + 1(0*1 - 1*1)\]

Step 3: Perform the Calculations

Calculate each term:- The first term: \( 1(1*1 - 0*1) = 1(1 - 0) = 1 \)- The second term: \( - 0(0*1 - 0*1) = 0 \)- The third term: \( + 1(0*1 - 1*1) = 1(0 - 1) = -1 \)Combine these results:\[det(A) = 1 + 0 - 1 = 0\]

Step 4: Conclude the Result

The calculated determinant of the matrix is 0. This tells us that the matrix does not have an inverse and is singular.

Key Concepts

Matrix Algebra

Matrix algebra is a specialized area of mathematics used to handle matrices, which are arrays of numbers organized in rows and columns. This branch of math is crucial in fields such as computer graphics, economics, and physics. Matrices can represent data, transformations, or systems of equations, making them versatile and powerful tools.

In matrix algebra, operations such as addition, subtraction, and multiplication are performed to manipulate these arrays. Addition and subtraction require matrices of the same dimensions, where corresponding elements are added or subtracted. Multiplication is a bit more complex, involving the dot product of rows and columns, which means that the number of columns in the first matrix must match the number of rows in the second matrix.

• Scalars: You can multiply matrices by scalar values, affecting each element in the matrix.
• Transpose: Flipping a matrix over its diagonal, switching row with column indices.

Manipulating matrices involves many techniques, including finding determinants, which play a central role in understanding matrix properties like invertibility.

3x3 Matrix Determinant

Finding the determinant of a 3x3 matrix is an essential skill in matrix algebra, as it provides insights into the matrix's properties, such as whether it has an inverse. For a matrix:- \[\begin{array}{ccc}a & b & c \d & e & f \g & h & i \\end{array}\]the determinant is computed using the formula: \[det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)\]Understanding this formula involves recognizing how you "expand" the determinant along a row or column, using the special pattern of cross-multiplying elements in the matrix:

• Each term involves a coefficient from the first row (\(a, b, c\)) and the minor determinant formed by omitting the coefficient's row and column.
• The signs alternate as you expand: the sequence is positive for the first, negative for the second, and positive again for the third term.
• The pattern of multiplication is intricate but systematic, allowing the determinant to encode essential mathematical information about the matrix.

Evaluating the determinant provides information such as whether the matrix is invertible, which is useful for solving systems of equations.

Singular Matrix

A singular matrix is one that does not have an inverse, making it uniquely important in matrix algebra. If the determinant of a matrix is zero, the matrix is classified as singular. This means it lacks the unique properties necessary to "reverse" its transformations or solve certain matrix equations.

Why is the determinant so crucial here?

• A zero determinant indicates that the matrix has no inverse, i.e., there is no matrix B such that \(AB = I\), where \(I\) is the identity matrix.
• Such matrices can result from linearly dependent rows or columns, signifying that the geometric transformation associated with the matrix compresses the space it operates over.
• In practical terms, solving linear systems represented by singular matrices often means no unique solutions exist—either they have none or an infinite number.

Recognizing singular matrices is critical in applications like numerical analysis, where the stability and solvability of systems must be assessed before further calculations.