Respuesta :
Answer:
The statements that are correct are:
c) [tex](in+a)(in+a^{-1})=2in+a+a^{-1}[/tex]
d) [tex]a^2b^7[/tex] is invertible.
e) [tex]a+b[/tex] is invertible.
Step-by-step explanation:
We are given that:
a and b are invertible n×n matrices.
We have to tell which of the following statements are true.
a)
[tex](ab)^{-1}=a^{-1}b^{-1}[/tex]
This statement is false.
Since:
[tex](ab)^{-1}=b^{-1}a^{-1}[/tex] and it may not be equal to the term [tex]a^{-1}b^{-1}[/tex]
b)
[tex]aba{-1}=b[/tex]
This expression could also be written as:
[tex]ab=ba[/tex]
Since on Post multiplying by a on both the sides.
But here we don't know whether the matrices are commutative or not.
Hence, the statement is false.
c)
[tex](in+a)(in+a^{-1})=2in+a+a^{-1}[/tex]
This statement is true.
since,
[tex](in+a)(in+a^{-1})=in(in+a^{-1})+a(in+a^{-1})\\\\=in^2+in.a^{-1}+a.in+aa^{-1}\\\\=in+a^{-1}+a+in\\\\=in+a^{-1}+a[/tex]
where in denote the identity matrix.
and we know that:
[tex]in^2=in[/tex]
d)
[tex]a^2b^7[/tex] is invertible.
This statement is true.
Since we know that prodct of invertible matrices is also invertible.
As [tex]a[/tex] is invertible so is [tex]a^2[/tex].
Also [tex]b[/tex] is invertible so is [tex]b^7[/tex].
Hence Product of [tex]a^2[/tex] and [tex]b^7[/tex] is also invertible.
i.e. [tex]a^2b^7[/tex] is invertible.
e)
[tex]a+b[/tex] is invertible.
This statement is true as sum of two invertible matrices is invertible.
f)
[tex](a+b).(a-b)=a^2-b^2[/tex]
This statement is false.
Since,
[tex](a+b).(a-b)=a(a-b)+b(a-b)\\\\=a.a-a.b+b.a-b.b\\\\=a^2-ab+ba-b^2[/tex]
Now as we are not given that:
[tex]ab=ba[/tex]
Hence, we could not say that:
[tex](a+b).(a-b)=a^2-b^2[/tex]
The correct statements which are true for all invertible [tex]\left({n\times n}\right)\cdot\left({n\times n}\right)[/tex] are:
(c). [tex]\left({in+a}\right)\left({in+{a^{-1}}}\right)=2in+a+{a^{-1}}[/tex]
(d). [tex]{a^2}{b^7}[/tex] is invertible.
(e). [tex]a+b[/tex] is invertible.
Further Explanation:
Given:
The matrix [tex]a[/tex] and [tex]b[/tex] are invertible [tex]\left({n\times n}\right)[/tex] matrices.
Calculation:
(a)
The statement is [tex]{\left({ab}\right)^{-1}}={a^{-1}}{b^{-1}}[/tex] false.
[tex]\begin{aligned}{\left({ab}\right)^{-1}}&={a^{-1}}{b^{-1}}\\\left({ab}\right){\left({ab}\right)^{-1}}&=\left({ab}\right){a^{-1}}{b^{-1}}\\I&\ne ab{a^{-1}}{b^{-1}}\\\end{aligned}[/tex]
The statement is [tex]{\left({ab}\right)^{-1}}={a^{-1}}{b^{-1}}[/tex] false.
(b)
The statement is [tex]ab{a^{-1}}=b[/tex].
Now multiply by a both the side.
[tex]\begin{aligned}ab{a^{-1}}a&=ba\\ab&\ne ba\\\end{aligned}[/tex]
The statement is [tex]ab{a^{-1}}=b[/tex] is false.
(c)
The statement is [tex]\left({in+a}\right)\left({in+{a^{-1}}}\right)=2in+a+{a^{-1}}[/tex].
Solve the above equation to check whether it is invertible.
[tex]\begin{aligned}\left({in+a}\right)\left({in+{a^{-1}}}\right)&=in\left({in+{a^{-1}}}\right)+a\left({in+{a^{-1}}}\right)\\&=i{n^2}+in\cdot{a^{-1}}+a\cdot in+a{a^{-1}}\\&=in+{a^{-1}}+a+in\\&=in+{a^{-1}}+a\\\end{aligned}[/tex]
The statement is true.
(d)
The statement is [tex]{a^2}{b^7}[/tex].
The product of invertible matrices is always invertible.
As [tex]a[/tex] is invertible so [tex]{a^2}[/tex] is also invertible.
As [tex]b[/tex] is invertible so [tex]{b^7}[/tex] is also invertible.
Hence, the product of [tex]{a^2}[/tex] and [tex]{b^7}[/tex] is also invertible.
The statement is true.
(e)
The statement [tex]a+b[/tex] is true as [tex]a+b[/tex] is always invertible.
(f)
The statement is [tex]\left({a+b}\right)\cdot\left({a-b}\right)={a^2}-{b^2}[/tex].
Solve the equation to check the inevitability.
[tex]\left({a+b}\right)\times\left({a-b}\right)={a^2}-ab+ba-{b^2}[/tex]
The statement is not true as [tex]ab=ba[/tex].
Hence, the correct statements which are true for all invertible [tex]\left({n\times n}\right)\cdot\left({n\times n}\right)[/tex] are:
(c). [tex]\left({in+a}\right)\left({in+{a^{-1}}}\right)=2in+a+{a^{-1}}[/tex]
(d). [tex]{a^2}{b^7}[/tex] is invertible.
(e). [tex]a+b[/tex] is invertible.
Learn more:
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3. Learn more about binomial and trinomial https://brainly.com/question/1394854
Answer details:
Grade: High School
Subject: Mathematics
Chapter: Linear equation
Keywords: Invertible, matrices, matrix, statement, function, true, determinants, elements, inverse.
