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I S. Y

A ,

S, N:

Q . INTRODUCTION TO DETERMINANTS

Specify whether the matrix has an inverse without trying to compute the inverse

−1 1 1 0 0

0 0 −1 0 0

0 0 1 −1 0

0 1 1 0 1

1−1 1 1 0

.

A

We use the deﬁnition of determinant. We calculate the determinant across the 2nd rows and 3rd column.

−1 1 1 0 0

0 0 −1 0 0

001−1 0

0 1 1 0 1

1−1 1 1 0

=−(−1)

−1 1 0 0

0 0 −1 0

0 1 0 1

1−1 1 0

=−(−1)

−(−1)

−1 1 0

0 1 1

1−1 0

=−(−1) (−(−1) (−(1)

−1 1

1−1

))

= (1)(1)(−1)((−1)(−1) −(1)(1))

= 0.

Since we have the determinant is 0, the matrix is NOT invertible.

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Q . THE PROPERTIES OF DETERMINANTS

(a) An n×nmatrix A is called skew-symmetric if At=−A. Show that if Ais skew-symmetric and nis

an odd positive integer, then Ais not invertible.

A

By the properties of determinant,

det(At) = det(−A)

det(A) = det(−A)

det(A) = (−1)ndet(A)

det(A) = −det(A).

So, we get det(A) = 0 which implies that Ais not invertible. Note that −Ameans that EVERY ROW of A

is multiplied by -1.

(b) Let A=

1λ0

111

001

. Determine those values of λfor which Ais invertible.

A

A=

1λ0

111

001

∼B=

1λ0

0 1 −λ1

0 0 1

.

By IMT, Ais invertible if and only if det(A) = det(B)̸= 0. Thus, Ais invertible if and only if 1−λ̸= 0 or

λ̸= 1.

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