The equation x 2 = 1 can be solved by setting x 2 − 1 = 0 and factoring the expression to obtain ( x − 1 ) ( x + 1 ) = 0 . This yields solutions x = 1 and x = − 1 . a) Using the factorization technique given above, what ( 2 × 2 ) matrix solutions do you obtain for the matrix equation X 2 = I ? b) Show that A = [ a 1 − a 2 1 − a ] is a solution to X 2 = I for every real number a . c) Let b = ± 1 . Show that B = [ b 0 c − b ] is a solution to X 2 = I for every real number c . d) Explain why the factorization technique used in part (a) did not yield all the solutions to the matrix equation X 2 = I .
The equation x 2 = 1 can be solved by setting x 2 − 1 = 0 and factoring the expression to obtain ( x − 1 ) ( x + 1 ) = 0 . This yields solutions x = 1 and x = − 1 . a) Using the factorization technique given above, what ( 2 × 2 ) matrix solutions do you obtain for the matrix equation X 2 = I ? b) Show that A = [ a 1 − a 2 1 − a ] is a solution to X 2 = I for every real number a . c) Let b = ± 1 . Show that B = [ b 0 c − b ] is a solution to X 2 = I for every real number c . d) Explain why the factorization technique used in part (a) did not yield all the solutions to the matrix equation X 2 = I .
Solution Summary: The author explains how the (2times 2) matrix solutions for the matrix equation
The equation
x
2
=
1
can be solved by setting
x
2
−
1
=
0
and factoring the expression to obtain
(
x
−
1
)
(
x
+
1
)
=
0
. This yields solutions
x
=
1
and
x
=
−
1
.
a) Using the factorization technique given above, what
(
2
×
2
)
matrix solutions do you obtain for the matrix equation
X
2
=
I
?
b) Show that
A
=
[
a
1
−
a
2
1
−
a
]
is a solution to
X
2
=
I
for every real number
a
.
c) Let
b
=
±
1
. Show that
B
=
[
b
0
c
−
b
]
is a solution to
X
2
=
I
for every real number
c
.
d) Explain why the factorization technique used in part (a) did not yield all the solutions to the matrix equation
X
2
=
I
.
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