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Chapter 16: Problem 37

Perform the indicated matrix multiplications. Using two rows and columns, show that \((-I)^{2}=I\).

Short Answer

Expert verified
The product \((-I)^{2}\) is indeed the identity matrix \( I \).

Step by step solution

01

Write Down the Identity Matrix

Let's write down the identity matrix, \( I \), for the square matrices of order 2:\[ I = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \]The identity matrix has 1s on the diagonal and 0s elsewhere.
02

Determine the Negative of the Identity Matrix

Find \( -I \) by multiplying each element of the identity matrix by -1. This gives us:\[ -I = \begin{pmatrix} -1 & 0 \ 0 & -1 \end{pmatrix} \]This negative identity matrix has -1s on the diagonal and 0s elsewhere.
03

Multiply the Negative Identity Matrix by Itself

Perform the matrix multiplication \( (-I) \times (-I) \):\[ \begin{pmatrix} -1 & 0 \ 0 & -1 \end{pmatrix} \times \begin{pmatrix} -1 & 0 \ 0 & -1 \end{pmatrix} = \begin{pmatrix} (-1)(-1) + (0)(0) & (-1)(0) + (0)(-1) \ (0)(-1) + (-1)(0) & (0)(0) + (-1)(-1) \end{pmatrix} \]Calculating each element:- First row, first column: \((-1)(-1) + (0)(0) = 1\)- First row, second column: \((-1)(0) + (0)(-1) = 0\)- Second row, first column: \((0)(-1) + (-1)(0) = 0\)- Second row, second column: \((0)(0) + (-1)(-1) = 1\)Thus, the resulting matrix is \( \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \), which is the identity matrix \( I \).
04

Verify the Result

Verify that the computed result matches the identity matrix \( I \). Indeed, the matrix \( \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \) is identical to the identity matrix of order 2.

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Key Concepts

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

Understanding the Identity Matrix
Identity matrix is a special type of square matrix. It is characterized by having 1s along its main diagonal and 0s in all other positions. For a matrix of order 2, it looks like this: \[ I = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \]
  • The identity matrix acts like the number 1 in arithmetic. When you multiply any matrix by the identity matrix, the original matrix remains unchanged.
  • This holds true for both left and right multiplication.
This concept applies to matrices of all sizes, not just 2x2. The identity matrix serves as a foundational tool in matrix operations.
Exploring the Negative Identity Matrix
The negative identity matrix is derived by multiplying every entry of the identity matrix by -1. For our 2x2 example, the negative identity matrix appears as follows:\[ -I = \begin{pmatrix} -1 & 0 \ 0 & -1 \end{pmatrix} \]
  • The diagonal elements become -1, while the rest of the elements remain 0.
  • This matrix is useful because it changes the sign of the elements in a matrix it is multiplied with, similar to multiplying by -1 in regular arithmetic.
Understanding the negative identity matrix helps us comprehend how matrices affect each other through multiplication.
Basics of Matrix Arithmetic
Matrix arithmetic involves operations such as addition, subtraction, and multiplication of matrices. Here鈥檚 a quick breakdown of these operations:
  • Addition: Matrices must be the same size. Add corresponding elements to get the new matrix.
  • Subtraction: Similar to addition. Subtract corresponding elements from each other.
  • Multiplication: Requires the number of columns in the first matrix to be equal to the number of rows in the second matrix.
  • The resulting matrix from multiplication has dimensions based on the outer dimensions of the two matrices being multiplied.
Understanding these basic operations is key to mastering matrix arithmetic and eventually more advanced topics.
Educational Algebra and Matrices
Educational algebra often introduces matrices as a tool for solving systems of equations. Matrices help simplify complex problems and present them in a structured format. Here鈥檚 how they integrate into algebra:
  • Matrices provide a method to represent coefficients in systems of equations, making it easier to visualize the relationships between variables.
  • Identity and negative identity matrices are essential in transforming and solving matrix equations.
  • Matrix operations, especially multiplication, are frequently used to solve equations efficiently.
By learning to manipulate matrices, students gain a valuable skillset that aids in understanding more intricate algebraic structures and problems.

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Most popular questions from this chapter

Perform the indicated matrix multiplications. Using Kirchhoff's laws on the circuit shown in Fig. 16.7 , the following matrix equation is found. By matrix multiplication, find the resulting system of equations. $$\left[\begin{array}{ccc}R_{1}+R_{2} & -R_{2} & 0 \\\\-R_{2} & R_{2}+R_{3}+R_{4} & -R_{4} \\ 0 & -R_{4} & R_{4}+R_{5}\end{array}\right]\left[\begin{array}{c}I_{1} \\\I_{2} \\\ I_{3}\end{array}\right]=\left[\begin{array}{c}V_{1} \\\0 \\\\-V_{2}\end{array}\right]$$

Solve the given problems. For matrices \(A=\left[\begin{array}{rr}2 & -4 \\ -1 & 3\end{array}\right], B=(A+1)^{-1},\) and \(C=A-I\) show that \(B C=C B\)

Solve the indicated systems of equations using the inverse of the coefficient matrix. It is necessary to set up the appropriate equations. Two batteries in an electric circuit have a combined voltage of 18 V, and one battery produces 6 V less than twice the other. What is the voltage of each?

Perform the indicated matrix multiplications. The path of an earth satellite can be written as $$\left[\begin{array}{ll}x & y\end{array}\right]\left[\begin{array}{rr}2.76 & -1 \\\1 & 2.81\end{array}\right]\left[\begin{array}{l}x \\\y\end{array}\right]=\left[7.76 \times 10^{7}\right]$$ where distances are in miles. What type of curve is represented? (See Section 14.1.)

Find \(B A^{-1}\). In Exercises \(32-34,\) find \(C A^{-1}\) $$B=\left[\begin{array}{ll}8 & -2 \\ 3 & 4\end{array}\right]$$ $$C=\left[\begin{array}{rrr}5 & -1 & 0 \\ 2 & -2 & 1 \\ -3 & 0 & 4\end{array}\right]$$ $$A=\left[\begin{array}{ll} 5 & -3 \\ 2 & -1 \end{array}\right]$$
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