Problem 71 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. Exploration Find a pair of \(3 \times 3\) matrices \(A\) and \(B\) to demonstrate that \(|A+B| \neq|A|+|B|\)

Step-by-Step Solution

Answer

The statement \(|A+B| \neq |A|+|B|\) is true. The determinant of the sum of matrices \(A\) and \(B\), which are chosen as the identity matrix and a matrix of ones respectively, is different from the sum of their individual determinants. Particularly, \(|A|+|B|\) is 1, while \(|A+B|\) equals 4.

Step 1: Selection of Matrices A and B

Choose two simple \(3x3\) matrices for \(A\) and \(B\). For instance, let's use \(A\) as the identity matrix and \(B\) as a matrix of ones. Hence, \[ A = \begin{bmatrix} 1&0&0 \ 0&1&0 \ 0&0&1 \end{bmatrix}, B = \begin{bmatrix} 1&1&1 \ 1&1&1 \ 1&1&1 \end{bmatrix} \]

Step 2: Calculation of Individual Determinants

Calculate the determinant of \(A\), denoted as \(|A|\), and the determinant of \(B\), denoted as \(|B|\). For \(|A|\), as it is an identity matrix, the determinant is 1. For \(|B|\), calculate by considering the elements of the first row and their respective cofactors. It results in:\\[ |B| = 1\cdot(det(\begin{bmatrix} 1&1 \ 1&1\end{bmatrix}))-1\cdot(det(\begin{bmatrix} 1&1 \ 1&1 \end{bmatrix}))+1\cdot(det(\begin{bmatrix} 1&1 \ 1&1 \end{bmatrix})) = 0 \]

Step 3: Calculation of Det(A+B)

Find the \((A+B)\) matrix first, then calculate its determinant, \(|(A+B)|\). The matrix \(A+B\) can be found by adding the corresponding elements of matrices \(A\) and \(B\). Hence, \[ A+B= \begin{bmatrix} 2&1&1\ 1&2&1\ 1&1&2\end{bmatrix}\]The determinant \(|(A+B)|\) is calculated in the same way as \(|B|\) and equals 4.

Step 4: Evaluation and Conclusion

From Steps 2 and 3, \(|A|+|B|\) equals 1 (as \(|B|\) is 0), while \(|A+B|\) equals 4. Clearly, \(|A+B| \neq |A|+|B|\). Therefore, the statement is true.

Key Concepts

3x3 Matrices

A matrix is a rectangular array of numbers arranged in rows and columns. When we talk about a 3x3 matrix, we mean a matrix with three rows and three columns. This is a square matrix, an important category in the world of linear algebra. Each element within a 3x3 matrix can be represented as an entry in a two-dimensional grid, often denoted by brackets. These matrices are used in various applications, including systems of equations, economics, and computer graphics.
To work effectively with 3x3 matrices, you need to know basic operations like addition, multiplication, and finding the determinant. Their size makes them complex enough to be useful in applications, yet simple enough to demonstrate clear matrix operations concepts.

Identity Matrix

The identity matrix is a special kind of matrix that acts like the number 1 when you're working with matrices. In a 3x3 identity matrix, all the diagonal elements (from the top left to the bottom right) are 1, and all other elements are 0. This can be written as: \[ I = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \]

• It doesn't change a matrix it's multiplied with.
• The determinant of an identity matrix is always 1, no matter its size.

In mathematics, an identity matrix is crucial because it serves as the multiplicative identity of matrix algebra, essentially leaving the other matrix unchanged in a multiplication operation.

Matrix Addition

Matrix addition involves adding two matrices by adding their corresponding elements together. For two matrices to be added, they must be of the same dimension, such as both being 3x3 matrices.
When adding two 3x3 matrices, for example:\[ A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{bmatrix},\hspace{5mm} B = \begin{bmatrix} b_{11} & b_{12} & b_{13} \ b_{21} & b_{22} & b_{23} \ b_{31} & b_{32} & b_{33} \end{bmatrix} \] The resulting matrix would be: \[ A + B = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} & a_{13} + b_{13} \ a_{21} + b_{21} & a_{22} + b_{22} & a_{23} + b_{23} \ a_{31} + b_{31} & a_{32} + b_{32} & a_{33} + b_{33} \end{bmatrix} \]
Note that the matrices must be 3x3 so that each element from one matrix can pair with an element from the other. This operation is fundamental in matrix mathematics and often used in more complex calculations.

Determinant Calculation

The determinant is a special number that can be calculated from a square matrix. It gives us key information about the matrix, like whether it's invertible and the volume scale factor. For a 3x3 matrix, the determinant is calculated using a more complex method than for 2x2 matrices, involving minors and cofactors.
For a matrix:\[ A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \] The determinant \(|A|\) is calculated as:\[ |A| = a(ei - fh) - b(di - fg) + c(dh - eg) \]

• A non-zero determinant indicates that the matrix is invertible.
• If the determinant is zero, the matrix doesn't have an inverse and is considered singular.

Understanding how to calculate the determinant of a 3x3 matrix is crucial, especially when solving systems of equations, performing matrix decompositions, or exploring linear transformations.