Given the matrix ( A = begin{bmatrix} 3 & 4 2 & 3 end{bmatrix} ), find the matrix ( B ) such that ( AB = I ), where ( I ) is the identity matrix.

Question

Given the matrix ( A = begin{bmatrix} 3 & 4  2 & 3 end{bmatrix} ), find the matrix ( B ) such that ( AB = I ), where ( I ) is the identity matrix.

Accepted Answer

Step:1 ：To find the matrix ( B ), we need to solve the equation ( AB = I ).Step:2 ：The identity matrix ( I ) is ( begin{bmatrix} 1 & 0  0 & 1 end{bmatrix} ).Step:3 ：Let ( B = begin{bmatrix} a & b  c & d end{bmatrix} ). Then, ( AB = begin{bmatrix} 3a+4c & 3b+4d  2a+3c & 2b+3d end{bmatrix} ).Step:4 ：Setting this equal to the identity matrix gives us the system of equations ( begin{cases} 3a+4c = 1  3b+4d = 0  2a+3c = 0  2b+3d = 1 end{cases} ).Step:5 ：Solving this system of equations gives ( a = 1/3  b = -1/2  c = -1/6  d = 2/3 ).Step:6 ：Therefore, ( B = begin{bmatrix} 1/3 & -1/2  -1/6 & 2/3 end{bmatrix} ).

Answer

Step: 1 ：To find the matrix ( B ), we need to solve the equation ( AB = I ).Step: 2 ：The identity matrix ( I ) is ( begin{bmatrix} 1 & 0  0 & 1 end{bmatrix} ).Step: 3 ：Let ( B = begin{bmatrix} a & b  c & d end{bmatrix} ). Then, ( AB = begin{bmatrix} 3a+4c & 3b+4d  2a+3c & 2b+3d end{bmatrix} ).Step: 4 ：Setting this equal to the identity matrix gives us the system of equations ( begin{cases} 3a+4c = 1  3b+4d = 0  2a+3c = 0  2b+3d = 1 end{cases} ).Step: 5 ：Solving this system of equations gives ( a = 1/3  b = -1/2  c = -1/6  d = 2/3 ).Step: 6 ：Therefore, ( B = begin{bmatrix} 1/3 & -1/2  -1/6 & 2/3 end{bmatrix} ).

Given the matrix $A = [\begin{matrix} 3 & 4 \\ 2 & 3 \end{matrix}]$ , find the matrix $B$ such that $A B = I$ , where $I$ is the identity matrix.

Answer

Popular Problems

Given the matrix A=[3423], find the matrix B such that AB=I, where I is the identity matrix.

Answer

Popular Problems

Given the matrix $A = [\begin{matrix} 3 & 4 \\ 2 & 3 \end{matrix}]$ , find the matrix $B$ such that $A B = I$ , where $I$ is the identity matrix.