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Chapter 8: Problem 97

Find square matrices \(A\) and \(B\) to demonstrate that \(|A+B| \neq|A|+|B|\).

Short Answer

Expert verified
The 2x2 identity matrices, \(A = \left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right]\) and \(B = \left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right]\), are viable examples. \(|A+B| = 4\) and \(|A|+|B| = 2\), therefore \(|A+B| \neq |A|+|B|\)

Step by step solution

01

Choose Matrices A and B

Choose two 2x2 matrices \(A\) and \(B\). Make sure to choose values for the elements of the matrices such that the determinant of their sum is not equal to the sum of their determinants. Perhaps the easiest choice is to just use 2x2 identity matrices for both \(A\) and \(B\), given by: \(A = \left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right]\) and \(B = \left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right]\)
02

Compute Determinants

Compute the determinants of \(A\), \(B\), and \(A+B\). For 2x2 matrices the determinant is computed as follows: \(|A| = a * d - b * c\), where \(a\), \(b\), \(c\), and \(d\) are the elements of the matrix. Given that both our matrices are identity matrices, this computation gives \(|A| = |B| = 1\). The matrix \(A+B\) is also an identity matrix with all elements doubled: \(A+B = \left[ \begin{array}{cc} 2 & 0 \\ 0 & 2 \end{array} \right]\), which means \(|A+B| = 4\).
03

Check the Equality of Determinants

Assess if \(|A+B| = |A|+|B|\). In this case, \(|A| = |B| = 1\), so \(|A|+|B| = 2\). However, \(|A+B| = 4\), so it is clear that \(|A+B| \neq |A|+|B|\), demonstrating the choice of \(A\) and \(B\) are valid examples to break the equality

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Matrices
A square matrix is a matrix with the same number of rows and columns. This balanced shape allows certain operations to work differently, laying a foundation for concepts like determinants. For a matrix to be square:
  • It must have equal dimensions, e.g., 2x2, 3x3, etc.
  • Square matrices can be subject to specific operations such as finding the determinant or eigenvalues.
When we work with square matrices, many properties emerge, such as the possibility of multiplication compatibility with other square matrices of the same size, and the potential for having an inverse matrix. Ensuring a matrix is square is a fundamental check before attempting these operations, as they generally aren't valid for non-square matrices.
Identity Matrix
An identity matrix is a special type of square matrix. In a 2x2 identity matrix, for instance, all the elements are zero except for the main diagonal elements, which are ones. It's like the number '1' in matrix form as it doesn鈥檛 change another matrix it multiplies:
  • The diagonal elements are all 1.
  • All other elements are 0.
For any matrix operation, the identity matrix behaves like a numerical "1"鈥攎ultiplying any matrix by the identity matrix leaves the original matrix unchanged. This is crucial when maintaining the value of original data in processes like matrix multiplication or trying to solve matrix equations.
2x2 Matrix
A 2x2 matrix is a simple and commonly used type of square matrix. With two rows and two columns, it offers entry points into the world of matrices:
  • The matrix typically looks like: \[\begin{bmatrix}a & b \c & d\end{bmatrix}\]
  • The determinant, a value indicating certain properties like invertibility, is calculated as \(|A| = ad - bc\).
2x2 matrices are often used in educational settings to demonstrate basic matrix operations, like addition or multiplication. Despite their small size, they hold significant mathematical potential, making them ideal for examples.
Matrix Addition
Matrix addition is the process of adding two matrices by adding their corresponding elements. If you have two matrices of the same size, you add them element by element:
  • All matrices involved must have the same dimensions.
  • It's done element-wise: if matrices A and B are added, each element of A is added to its corresponding element in B.
When working with square matrices, especially during addition, it鈥檚 crucial to check if the dimensions are equal to ensure a valid operation. While it might seem straightforward, matrix addition can highlight interesting phenomena, such as when the determinant of the resulting matrix differs from the sum of the determinants of the original matrices, showing that addition affects a matrix more complexly than mere element-wise summation would suggest.

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Most popular questions from this chapter

Use determinants to find the area of a triangle with vertices \((3,-1),(7,-1),\) and (7,5) Confirm your answer by plotting the points in a coordinate plane and using the formula Area \(=\frac{1}{2}(\text { base })(\text { height })\).

Evaluate the determinant(s) to verify the equation. $$\left|\begin{array}{ll} w & x \\ y & z \end{array}\right|=\left|\begin{array}{ll} w & x+c w \\ y & z+c y \end{array}\right|$$

Write a cryptogram for the message using the matrix \(A=\left[\begin{array}{rrr} 1 & 2 & 2 \\ 3 & 7 & 9 \\ -1 & -4 & -7 \end{array}\right]\). OPERATION OVERLOAD

A video of the path of a ball thrown by a baseball player was analyzed with a grid covering the TV screen. The tape was paused three times, and the position of the ball was measured each time. The coordinates obtained are shown in the table. \((x \text { and } y\) are measured in feet.) $$\begin{array}{|l|c|c|c|} \hline \text { Horizontal Distance, } x & 0 & 15 & 30 \\ \hline \text { Height, } y & 5.0 & 9.6 & 12.4 \\ \hline \end{array}$$ (a) Use a system of equations to find the equation of the parabola \(y=a x^{2}+b x+c\) that passes through the three points. Solve the system using matrices. (b) Use a graphing utility to graph the parabola. (c) Graphically approximate the maximum height of the ball and the point at which the ball struck the ground. (d) Analytically find the maximum height of the ball and the point at which the ball struck the ground. (e) Compare your results from parts (c) and (d).

Use the matrix capabilities of a graphing utility to evaluate the determinant. $$\left|\begin{array}{rrrr} 0 & -3 & 8 & 2 \\ 8 & 1 & -1 & 6 \\ -4 & 6 & 0 & 9 \\ -7 & 0 & 0 & 14 \end{array}\right|$$
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