91Ó°ÊÓ

First, write down the given matrix A: \(A = \begin{pmatrix} 2 & 1 \\ -2 & -1 \end{pmatrix}\) Now, let's assume that the matrix B is in the following form: \(B = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\) We need to find the values of a, b, c, and d such that B^2 = A. So, we will perform the matrix multiplication B * B and equate the resulting matrix to A. \(B * B = \begin{pmatrix} a^2 + bc & ab+bd \\ ac+cd & bc+d^2 \end{pmatrix}\) Now equate B * B to A, and we get: \(\begin{pmatrix} a^2 + bc & ab+bd \\ ac+cd & bc+d^2 \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ -2 & -1 \end{pmatrix}\) From this equation, we can write down the following system of equations: 1. a^2 + bc = 2 2. ab + bd = 1 3. ac + cd = -2 4. bc + d^2 = -1 From equation 1, we have: \(bc = 2 - a^2 \) From equation 4, we have: \(bc = -1 - d^2 \) Now set both equations equal to each other: \(-1 - d^2 = 2 - a^2 \) We can solve this equation to obtain a and d values: Let a=1 and d=-1 Now, substituting a and d in equation 2 and 3, we can find the values for b and c. 2. b(1) - b = 1 => b = -1 3. c - c = -2 => c = 1 Now we have all the values to form matrix B: \(B = \begin{pmatrix} 1 & -1 \\ 1 & -1 \end{pmatrix}\)

First, write down the given matrix A: \(A = \begin{pmatrix} 9 & -5 & 3 \\ 0 & 4 & 3 \\ 0 & 0 & 1 \end{pmatrix}\) Now, let's assume that the matrix B is in the following form: \(B = \begin{pmatrix} e & f & g \\ 0 & h & i \\ 0 & 0 & j \end{pmatrix}\) To find the values of e, f, g, h, i, and j such that B^2 = A, perform the matrix multiplication B * B and equate the resulting matrix to A. \(B * B = \begin{pmatrix} e^2 & ef+fh & eg+gi+j^2 \\ 0 & h^2 & hi \\ 0 & 0 & j^2 \end{pmatrix}\) Now equate B * B to A, which gives us: \(\begin{pmatrix} e^2 & ef+fh & eg+gi+j^2 \\ 0 & h^2 & hi \\ 0 & 0 & j^2 \end{pmatrix} = \begin{pmatrix} 9 & -5 & 3 \\ 0 & 4 & 3 \\ 0 & 0 & 1 \end{pmatrix}\) From this equation, we can write down the following system of equations: 1. \(e^2 = 9 \) 2. ef + fh = -5 3. eg + gi + j^2 = 3 4. h^2 = 4 5. hi = 3 6. j^2 = 1 We can solve these equations one by one. 1. e = ±3 4. h = ±2 6. j = ±1 We can see that for the B matrix to make sense in this context, we should choose the positive values of e, h, and j. So, we have: e = 3, h = 2, and j = 1 Now we can use equations 2 and 5 to find the remaining variables, f and i. 2. 3f + 2f = -5 => f = -1 5. 2i = 3 => i = 1.5 Now we have all the values to form matrix B: \(B = \begin{pmatrix} 3 & -1 & 0 \\ 0 & 2 & 1.5 \\ 0 & 0 & 1 \end{pmatrix}\)