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Chapter 2: Q 40E (page 98)

Question:Show that if a square matrix Ahas two equal columns, thenA is not invertible.

Short Answer

Expert verified

When a matrix has equal columns then rrefA鈮�I3, so, the matrix is not invertible.

Step by step solution

01

Consider the matrix

Consider the matrix with two equal columns.

A=abbcddeff

The matrix is said to be invertible if and only if the determinant of the matrix is nonzero.

The matrix is said to be invertible, if and only if, rref(A)=I3.

02

Perform the elementary row operation

Consider the matrix,

A=abbcddeff

The reduced matrix is,

abbcddeff=cdd0cf鈭�edccf鈭�edc000=100011000鈮�I3

As rrefA鈮�I3, so, the matrix is non-invertible.

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Most popular questions from this chapter

Question: For two invertible n 脳 n matrices Aand B, determine which of the formulas stated in Exercise 67 through 75 are necessarily true.

(A鈭�B)(A+B)=A2鈭�B2

For which values of the constant k is the following matrix invertible?

[11112k14k2]

TRUE OR FALSE?

The formula rref(AB) =rref(A)rref(B)holds for alln脳p matrices A and for allp脳m matrices.

There exists an invertible 2 脳 2 matrix A such that A-1=[1111].

Consider the circular face in the accompanying figure. For each of the matrices A in Exercises 24 through 30, draw a sketch showing the effect of the linear transformation T(x)鈫�=Ax鈫�on this face.

26.[0110]
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