What is the value of element 1,1 in matrix $X$?
$\left.\begin{array}{ll}1 & 2 \\ 0 & m\end{array}\right] X=\left[\begin{array}{c}m \\ 2 m\end{array}\right]$
A. $m-4$
B. $m+4$
C. $m$
D. 2
Answer (2)
Set up the matrix equation A X = B with X = [ x 11 x 21 ] .
Perform the matrix multiplication to obtain a system of equations.
Solve the system of equations for x 11 , assuming m = 0 .
The value of element 1,1 in matrix X is m − 4 .
Explanation
Understanding the Problem We are given the matrix equation A X = B , where A = [ 1 0 2 m ] , X is a 2 × 1 matrix, and B = [ m 2 m ] . We want to find the value of the element in the first row and first column of matrix X , which we denote as x 11 .
Setting up the Matrix Multiplication Let X = [ x 11 x 21 ] . Then A X = [ 1 0 2 m ] [ x 11 x 21 ] = [ x 11 + 2 x 21 m x 21 ] .
Forming the System of Equations We are given that A X = B = [ m 2 m ] . Therefore, we have the system of equations:
x 11 + 2 x 21 = m and m x 21 = 2 m .
Solving for x_{21} From the second equation, if m = 0 , then we can divide both sides by m to get x 21 = 2 .
Solving for x_{11} Substituting x 21 = 2 into the first equation, we get x 11 + 2 ( 2 ) = m , so x 11 + 4 = m . Solving for x 11 , we find x 11 = m − 4 .
Considering the Case m=0 If m = 0 , then A = [ 1 0 2 0 ] and B = [ 0 0 ] . The equation A X = B becomes [ 1 0 2 0 ] [ x 11 x 21 ] = [ 0 0 ] , which gives x 11 + 2 x 21 = 0 . In this case, x 11 = − 2 x 21 . Since x 21 can be any real number, x 11 can take infinitely many values. However, if we assume m = 0 , then x 11 = m − 4 .
Final Answer Therefore, assuming m = 0 , the value of element 1,1 in matrix X is m − 4 .
Examples
Matrix equations are used in computer graphics to perform transformations on objects. For example, the matrix A could represent a scaling or rotation, the matrix X could represent the coordinates of a point, and the matrix B could represent the transformed coordinates of the point. Solving the equation A X = B allows us to find the original coordinates X given the transformed coordinates B and the transformation matrix A . This is essential for rendering 3D scenes and creating animations.
The value of element 1,1 in the matrix X is m − 4 . This was derived by solving the system of equations formed from the matrix multiplication. Therefore, the correct option is A. m − 4 .
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