
#1
Sep1305, 02:06 PM

P: 20

(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. 



#2
Sep1305, 03:13 PM

HW Helper
P: 1,024

[tex] \left( {\begin{array}{*{20}c} 2 & 0 \\ 4 & 1 \\ \end{array} } \right) \cdot \left( {\begin{array}{*{20}c} 2 & 0 \\ 4 & 1 \\ \end{array} } \right) \cdot \left( {\begin{array}{*{20}c} 2 & 0 \\ 4 & 1 \\ \end{array} } \right) [/tex] 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. [tex] \left( {\begin{array}{*{20}c} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \\ \end{array} } \right)^n = \left( {\begin{array}{*{20}c} {a^n } & 0 & 0 \\ 0 & {b^n } & 0 \\ 0 & 0 & {c^n } \\ \end{array} } \right) [/tex] For 1e, simply follow the instructions. I assume the constant will have to be multiplied with the unity matrix. [tex] f\left( A \right) = A^2  5A + 2 = \left( {\begin{array}{*{20}c} 2 & 0 \\ 4 & 1 \\ \end{array} } \right)^2  5\left( {\begin{array}{*{20}c} 2 & 0 \\ 4 & 1 \\ \end{array} } \right) + 2\left( {\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ \end{array} } \right) [/tex] 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. [tex] \left( {\begin{array}{*{20}c} 1 & 3 \\ 0 & 1 \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} a & b \\ c & d \\ \end{array} } \right)^2 = \left( {\begin{array}{*{20}c} a & b \\ c & d \\ \end{array} } \right) \cdot \left( {\begin{array}{*{20}c} a & b \\ c & d \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {a^2 + bc} & {ab + bd} \\ {ac + cd} & {bc + d^2 } \\ \end{array} } \right) [/tex] 



#3
Sep1305, 03:19 PM

Sci Advisor
HW Helper
P: 1,996

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. 



#4
Sep1305, 04:23 PM

P: 20

Matrix Questions 



#5
Dec1209, 11:06 AM

P: 1

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


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