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Chapter 6: Q1E (page 289)

Use Gaussian elimination to find the determinant of the matrices A in Exercises 1 through 10.

1.[111133225]

Short Answer

Expert verified

Therefore, the determinant of given matrix is given by,

detA=6

Step by step solution

01

Definition

Gaussian elimination method is used to solve a system of linear equations.


Gaussian elimination provides a relatively efficient way of constructing the inverse to a matrix.


Interchanging two rows, multiplying a row by a constant (any constant which is not zero).

02

Given

Given Matrix,

A=111133225

03

To find determinant by using Gaussian Eliminations

First, we multiply the first row by -1 and add it to the second row. Then, we multiply the first row by -2 and add it to the third row.

We get,

A1=111022003

We had zero row swaps, so

detA=(-1)0|A1|=(-1)0.1.2.3=6

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Most popular questions from this chapter

(For those who have studied multivariable calculus.) Let Tbe an invertible linear transformation from鈩�2to鈩�2, represented by the matrix M. Let惟1be the unit square in 鈩�2and惟2its image under T . Consider a continuous functionf(x,y)from鈩�2to鈩�, and define the functiong(u,v)=f(T(u,v)). What is the relationship between the following two double integrals?

鈭�惟2f(x,y)dAand鈭�惟1g(u,v)dA

Your answer will involve the matrix M. Hint: What happens whenf(x,y)=1, for allx,y?

If an n脳n matrixAis invertible, then there must be an (n-1)脳(n-1)sub matrix of(obtained by deleting a row and a column of) that is invertible as well.

Consider two vectorsv鈬赌 and w鈬赌in鈩�3. Form the matrix A=[v鈬赌xw鈬赌v鈬赌w鈬赌] . Express detA in terms of||v鈬赌xw鈬赌||. For which choices of v鈬赌 and w鈬赌 is Ainvertible?

Consider the function F(A)=F[v鈫�w鈫�]=v鈫�脳w鈫�from 鈩�2脳2 to鈩�, the dot product of the column vectors of A.
a. Is Flinear in both columns of A? See Example 6.
b. Is F linear in both rows of A?
c. Is Falternating on the columns of A? See Example 4.

If a square matrix A is invertible, then its classical adjoint adj(A) is invertible as well.
See all solutions

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