Matrix Questions: Solve A^3, C^2003, f(A), e^C & Square Roots of A

[niteshadw]

(1)
Let
A =
2 0
4 1

B =
2 0 −4
3 −2 6

C =
5 0 0
0 −1 0
0 0 0

and let f(t) = t^2 - 5t + 2. Compute the following if possible.

(a) A^3
(b) C^2003
(e) f(A)
(g) We define the matrix exponential by the Taylor series:

e^C = I + C + 1/2! * C^2 + 1/3! * C^3 + · · · + 1/n! * Cn + · · · .

Calculate e^C

(2)
An n × n matrix S (with real entries) is called a square root of the n × n matrix A (with real entries), if S2 = A.

Find the square roots of the matrix
A=
1 3
0 1

========================

I don't have an idea on how to do the problems just posted, I can do the rest and those that I did not post, but I never learned #2 and I don't know how to take powers of matrix nor recall series. Would anyone be kind enough to explain how to do these problems. It would be very much appreciated. Thank you.

 

[TD]

(1)
(a) A^3
(b) C^2003
(e) f(A)
(g) We define the matrix exponential by the Taylor series:

e^C = I + C + 1/2! * C^2 + 1/3! * C^3 + · · · + 1/n! * Cn + · · · .

Calculate e^C

(2)
An n × n matrix S (with real entries) is called a square root of the n × n matrix A (with real entries), if S2 = A.

Find the square roots of the matrix
A=
1 3
0 1

========================

I don't have an idea on how to do the problems just posted, I can do the rest and those that I did not post, but I never learned #2 and I don't know how to take powers of matrix nor recall series. Would anyone be kind enough to explain how to do these problems. It would be very much appreciated. Thank you.

Well if you know matrix multiplication, 1a shouldn't be a problem. Calculate:

<br /> \left( {\begin{array}{*{20}c}<br /> 2 &amp; 0 \\<br /> 4 &amp; 1 \\<br /> <br /> \end{array} } \right) \cdot \left( {\begin{array}{*{20}c}<br /> 2 &amp; 0 \\<br /> 4 &amp; 1 \\<br /> <br /> \end{array} } \right) \cdot \left( {\begin{array}{*{20}c}<br /> 2 &amp; 0 \\<br /> 4 &amp; 1 \\<br /> <br /> \end{array} } \right)<br />

The power 2003 seems horrible, but C is a diagonal matrix and that has a handy property for powers. This should make 1b and 1g possible.

<br /> \left( {\begin{array}{*{20}c}<br /> a &amp; 0 &amp; 0 \\<br /> 0 &amp; b &amp; 0 \\<br /> 0 &amp; 0 &amp; c \\<br /> <br /> \end{array} } \right)^n = \left( {\begin{array}{*{20}c}<br /> {a^n } &amp; 0 &amp; 0 \\<br /> 0 &amp; {b^n } &amp; 0 \\<br /> 0 &amp; 0 &amp; {c^n } \\<br /> <br /> \end{array} } \right)<br />

For 1e, simply follow the instructions. I assume the constant will have to be multiplied with the unity matrix.

<br /> f\left( A \right) = A^2 - 5A + 2 = \left( {\begin{array}{*{20}c}<br /> 2 &amp; 0 \\<br /> 4 &amp; 1 \\<br /> <br /> \end{array} } \right)^2 - 5\left( {\begin{array}{*{20}c}<br /> 2 &amp; 0 \\<br /> 4 &amp; 1 \\<br /> <br /> \end{array} } \right) + 2\left( {\begin{array}{*{20}c}<br /> 1 &amp; 0 \\<br /> 0 &amp; 1 \\<br /> <br /> \end{array} } \right)<br />

For 2, you know you have to start with a 2x2 matrix. Take a general one, take the square and identify the elements. This will give a fairly easy system.

<br /> \left( {\begin{array}{*{20}c}<br /> 1 &amp; 3 \\<br /> 0 &amp; 1 \\<br /> <br /> \end{array} } \right) = \left( {\begin{array}{*{20}c}<br /> a &amp; b \\<br /> c &amp; d \\<br /> <br /> \end{array} } \right)^2 = \left( {\begin{array}{*{20}c}<br /> a &amp; b \\<br /> c &amp; d \\<br /> <br /> \end{array} } \right) \cdot \left( {\begin{array}{*{20}c}<br /> a &amp; b \\<br /> c &amp; d \\<br /> <br /> \end{array} } \right) = \left( {\begin{array}{*{20}c}<br /> {a^2 + bc} &amp; {ab + bd} \\<br /> {ac + cd} &amp; {bc + d^2 } \\<br /> <br /> \end{array} } \right)<br />

 

[shmoe]

You know how to multiply matrices together right? That's all powers are, A^2=AxA, A^3=AxAxA, etc. (the little "x" meaning "times" here)

For the huge power C^2003, find C^2, C^3, C^4, ... as many as you need to until you see a pattern. You should be able to write a nice general expression for C^n, which will help for part (g).

For part (g), use your expression for C^n to write each entry of e^C as an infinite sum. The usual series for e^x where x is a real number is the same as the one they've given for e^C with C's and x's interchanged, so you should be able to write e^C in a nice form using this.


For the square root question, this will seem like a lame suggestion but it doesn't look like you're expected to know a general method to compute the square root of a matrix (when it exists). So try to guess an S that works here. Maybe computing some powers of A will give you some inspiration.

 

[niteshadw]

Well if you know matrix multiplication, 1a shouldn't be a problem. Calculate:

<br /> \left( {\begin{array}{*{20}c}<br /> 2 &amp; 0 \\<br /> 4 &amp; 1 \\<br /> <br /> \end{array} } \right) \cdot \left( {\begin{array}{*{20}c}<br /> 2 &amp; 0 \\<br /> 4 &amp; 1 \\<br /> <br /> \end{array} } \right) \cdot \left( {\begin{array}{*{20}c}<br /> 2 &amp; 0 \\<br /> 4 &amp; 1 \\<br /> <br /> \end{array} } \right)<br />

The power 2003 seems horrible, but C is a diagonal matrix and that has a handy property for powers. This should make 1b and 1g possible.

<br /> \left( {\begin{array}{*{20}c}<br /> a &amp; 0 &amp; 0 \\<br /> 0 &amp; b &amp; 0 \\<br /> 0 &amp; 0 &amp; c \\<br /> <br /> \end{array} } \right)^n = \left( {\begin{array}{*{20}c}<br /> {a^n } &amp; 0 &amp; 0 \\<br /> 0 &amp; {b^n } &amp; 0 \\<br /> 0 &amp; 0 &amp; {c^n } \\<br /> <br /> \end{array} } \right)<br />

For 1e, simply follow the instructions. I assume the constant will have to be multiplied with the unity matrix.

<br /> f\left( A \right) = A^2 - 5A + 2 = \left( {\begin{array}{*{20}c}<br /> 2 &amp; 0 \\<br /> 4 &amp; 1 \\<br /> <br /> \end{array} } \right)^2 - 5\left( {\begin{array}{*{20}c}<br /> 2 &amp; 0 \\<br /> 4 &amp; 1 \\<br /> <br /> \end{array} } \right) + 2\left( {\begin{array}{*{20}c}<br /> 1 &amp; 0 \\<br /> 0 &amp; 1 \\<br /> <br /> \end{array} } \right)<br />

For 2, you know you have to start with a 2x2 matrix. Take a general one, take the square and identify the elements. This will give a fairly easy system.

<br /> \left( {\begin{array}{*{20}c}<br /> 1 &amp; 3 \\<br /> 0 &amp; 1 \\<br /> <br /> \end{array} } \right) = \left( {\begin{array}{*{20}c}<br /> a &amp; b \\<br /> c &amp; d \\<br /> <br /> \end{array} } \right)^2 = \left( {\begin{array}{*{20}c}<br /> a &amp; b \\<br /> c &amp; d \\<br /> <br /> \end{array} } \right) \cdot \left( {\begin{array}{*{20}c}<br /> a &amp; b \\<br /> c &amp; d \\<br /> <br /> \end{array} } \right) = \left( {\begin{array}{*{20}c}<br /> {a^2 + bc} &amp; {ab + bd} \\<br /> {ac + cd} &amp; {bc + d^2 } \\<br /> <br /> \end{array} } \right)<br />

Thank you very much for the explanation. I admit the A^3 wa a pretty bad question, I was concentrating on the power to 2003; but I did that and now thanks to your help I'll hope to do 2003. The professor did not manage to mention any of these explanations - we don't even have a book for linear but instead notes...

For part (g), use your expression for C^n to write each entry of e^C as an infinite sum. The usual series for e^x where x is a real number is the same as the one they've given for e^C with C's and x's interchanged, so you should be able to write e^C in a nice form using this.


For the square root question, this will seem like a lame suggestion but it doesn't look like you're expected to know a general method to compute the square root of a matrix (when it exists). So try to guess an S that works here. Maybe computing some powers of A will give you some inspiration.

Thanks, I'll give it a try and no, I never seen anything that dealt with the square root of a matrix...

 

[arthdt]

how to do the square of 3×3 matrix?
please reply me with full method