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Chapter 9: Problem 84

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Two \(2 \times 2\) invertible matrices can have a matrix sum that is not invertible.

Short Answer

Expert verified
The statement 'Two \(2 \times 2\) invertible matrices can have a matrix sum that is not invertible' is conditionally true. It is not true or false universally, as it depends on the specific matrices.

Step by step solution

01

Definition of Invertible Matrices

An invertible matrix (also known as a non-singular matrix or a non-degenerate matrix) is a square matrix that has an inverse. In other words, if \(A\) is a matrix, there exists a matrix \(A^{-1}\) such that \(AA^{-1} = A^{-1}A = I\), where \(I\) is the identity matrix of the same order.
02

Adding Invertible Matrices

Let's denote our two \(2 \times 2\) invertible matrices as \(A\) and \(B\). Their sum is represented as \(C=A+B\). The elements of \(C\) will be the sum of corresponding elements of \(A\) and \(B\).
03

Checking Invertibility of the Sum

The statement given implies that the sum of two invertible matrices, \(C=A+B\), is not invertible. However, this is not necessarily true, because the sum of two invertible matrices can still be invertible, or it may not be invertible. It actually depends on the specific matrices \(A\) and \(B\). Therefore, the statement is not definitively true or false as it depends on the specific matrices. It's important to note, however, that generally speaking, the operations applied to matrices, such as addition in this case, do not necessarily preserve the property of invertibility.

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Most popular questions from this chapter

What is meant by the order of a matrix? Give an example with your explanation.

The figure shows the letter \(L\) in a rectangular coordinate system. The figure can be represented by the matrix $$ B=\left[\begin{array}{llllll} {0} & {3} & {3} & {1} & {1} & {0} \\ {0} & {0} & {1} & {1} & {5} & {5} \end{array}\right] $$ Each column in the matrix describes a point on the letter. The order of the columns shows the direction in which a pencil must move to draw the letter. The L is completed by connecting the last point in the matrix, (0,5), to the starting point, (0,0) . Use these ideas to solve Exercises 53-60. a. If \(A=\left[\begin{array}{rr}{1} & {0} \\ {0} & {-1}\end{array}\right],\) find \(A B\). b. Graph the object represented by matrix \(A B .\) What effect does the matrix multiplication have on the letter \(L\) represented by matrix \(B ?\)

Perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows. If an operation is not defined, state the reason. $$ A=\left[\begin{array}{rr} {4} & {0} \\ {-3} & {5} \\ {0} & {1} \end{array}\right] \quad B=\left[\begin{array}{rr} {5} & {1} \\ {-2} & {-2} \end{array}\right] \quad C=\left[\begin{array}{rr} {1} & {-1} \\ {-1} & {1} \end{array}\right] $$ $$ 4 B-3 C $$

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{array}{l} {2 x+2 y+7 z=-1} \\ {2 x+y+2 z=2} \\ {4 x+6 y+z=15} \end{array}\right. $$

Find (if possible) the following matrices: a. \(A B\) b. \(B A\) $$ A=\left[\begin{array}{llll} {1} & {2} & {3} & {4} \end{array}\right], \quad B=\left[\begin{array}{l} {1} \\ {2} \\ {3} \\ {4} \end{array}\right] $$
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