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Chapter 9: Problem 2

Let \(A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\) and \(I_{2}=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] .\) Show that \(A I_{2}=I_{2} A=A\), thus proving that \(I_{2}\) is the identity element for matrix multiplication for \(2 \times 2\) square matrices.

Short Answer

Expert verified
The identity matrix \(I_2\) satisfies \(AI_2 = I_2A = A\), confirming it is the identity element for 2 x 2 matrices.

Step by step solution

01

Define Matrix A and Identity Matrix

Matrix A is given by \[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \] The identity matrix for 2x2 matrices is \[ I_2 = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \]
02

Multiply Matrix A by Identity Matrix

Perform the multiplication of matrix A by the identity matrix: \[ AI_2 = \begin{bmatrix} a & b \ c & d \end{bmatrix} \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \]Multiply each element of the rows of A by the corresponding columns of \(I_2\):
03

Calculate Elements of Resultant Matrix

For each element in the resulting matrix: \[ \begin{bmatrix} a \times 1 + b \times 0 & a \times 0 + b \times 1 \ c \times 1 + d \times 0 & c \times 0 + d \times 1 \end{bmatrix} = \begin{bmatrix} a & b \ c & d \end{bmatrix} = A \]
04

Multiply Identity Matrix by Matrix A

Perform the multiplication of the identity matrix by matrix A: \[ I_2 A = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \begin{bmatrix} a & b \ c & d \end{bmatrix} \]Multiply each element of the rows of \(I_2\) by the corresponding columns of A:
05

Calculate Elements of Resultant Matrix

For each element in the resulting matrix: \[ \begin{bmatrix} 1 \times a + 0 \times c & 1 \times b + 0 \times d \ 0 \times a + 1 \times c & 0 \times b + 1 \times d \end{bmatrix} = \begin{bmatrix} a & b \ c & d \end{bmatrix} = A \]
06

Conclude Identity Property

Since both multiplications \(AI_2\) and \(I_2A\) resulted in matrix A, it is proven that \(I_2\) is the identity element for matrix multiplication for 2 x 2 matrices: \[ AI_2 = I_2A = A \]

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Key Concepts

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

2x2 Matrices
A 2x2 matrix is a rectangular array of numbers arranged in two rows and two columns. For example, \[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \] is a 2x2 matrix. Here, \(a\), \(b\), \(c\), and \(d\) are the elements of the matrix. These matrices are used in various mathematical contexts, such as solving systems of linear equations, transformations in geometry, and more.

  • The first row is \([a, b]\).
  • The second row is \([c, d]\).
  • Similarly, the first column is \([a, c]\).
  • The second column is \([b, d]\).
Matrix Multiplication
Matrix multiplication involves multiplying the elements of the rows of the first matrix by the elements of the columns of the second matrix and summing them up. For example, if we have two matrices \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\) and \(B = \begin{bmatrix} w & x \ y & z \end{bmatrix}\), then their product is calculated as follows:
\[ AB = \begin{bmatrix} (aw + by) & (ax + bz) \ (cw + dy) & (cx + dz) \end{bmatrix}\]
  • First, multiply the elements across rows of \(A\) with columns of \(B\).
  • Then, sum the products to get each element of the resulting matrix.
  • Repeat this process for all elements.
Always remember that matrix multiplication is not commutative; \(AB eq BA\) in general.
Identity Matrix
An identity matrix is a special type of square matrix with ones on the main diagonal and zeros elsewhere. For a 2x2 matrix, the identity matrix \(I\textsubscript{2}\) is given by: \[ I_2 = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \] 鈥� This matrix acts as the multiplicative identity in matrix operations, much like the number 1 does in regular multiplication.
  • For any 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), multiplying by \(I\textsubscript{2}\) will not change it: \(AI\textsubscript{2} = I\textsubscript{2}A = A\).
  • This was proven in the original exercise by multiplying matrix \(A\) with \(I\textsubscript{2}\) from both the left and right, showing that the resultant matrix remains the same as \(A\).
This confirms the important property of the identity matrix.

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Most popular questions from this chapter

For each pair of matrices \(A\) and \(B,\) find \((a) A B\) and \((b) B A\). $$A=\left[\begin{array}{rrr} -1 & 0 & 1 \\ 0 & 1 & 1 \\ -1 & -1 & 0 \end{array}\right], B=\left[\begin{array}{lll} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right]$$

Given \(A=\left[\begin{array}{rr}4 & -2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{rr}5 & 1 \\ 0 & -2 \\ 3 & 7\end{array}\right],\) and \(C=\left[\begin{array}{rrr}-5 & 4 & 1 \\ 0 & 3 & 6\end{array}\right],\) find each product when possible. $$A^{2}$$

Use the determinant theorems to find the value of each determinant. $$\left|\begin{array}{rrrr} 4 & 0 & 0 & 2 \\ -1 & 0 & 3 & 0 \\ 2 & 4 & 0 & 1 \\ 0 & 0 & 1 & 2 \end{array}\right|$$

Find the equation of the circle passing through the given points. $$(-1,3),(6,2), \text { and }(-2,-4)$$

Solve each problem. Tire Sales The number of automobile tire sales is dependent on several variables. In one study the relationship among annual tire sales \(S\) (in thousands of dollars), automobile registrations \(R\) (in millions), and personal disposable income \(I\) (in millions of dollars) was investigated. The results for three years are given in the table. To describe the relationship among these variables, we can use the equation $$ S=a+b R+c l $$ where the coefficients \(a, b,\) and \(c\) are constants that must be determined before the equation can be used. (Source: Jarrett, J., Business Forecasting Methods, Basil Blackwell, Ltd.) (a) Substitute the values for \(S, R,\) and \(I\) for each year from the table into the equation \(S=a+b R+c I,\) and obtain three linear equations involving \(a, b,\) and \(c\) (b) Use a graphing calculator to solve this linear system for \(a, b,\) and \(c .\) Use matrix inverse methods. (c) Write the equation for \(S\) using these values for the coefficients. (d) If \(R=117.6\) and \(I=310.73,\) predict \(S .\) (The actual value for \(S\) was \(11,314 .\) ) (e) If \(R=143.8 \text { and } I=829.06, \text { predict } S . \text { (The actual value for } S \text { was } 18,481 .)\) $$\begin{array}{|c|c|c|} \hline S & R & I \\ \hline 10,170 & 112.9 & 307.5 \\\ \hline 15,305 & 132.9 & 621.63 \\ \hline 21,289 & 155.2 & 1937.13 \\\ \hline \end{array}$$
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