If \({\rm{A}} = \left[ {\begin{array}{*{20}{c}} 1&0&{ - 2}\\ 2&{ - 3}&4 \end{array}} \right]\) , then the matrix X for which 2X + 3A = 0 holds true is
Solving for Matrix X in the Equation 2X + 3A = 0
We are given a specific matrix \({\rm{A}}\) and need to find another matrix \({\rm{X}}\) that satisfies a given matrix equation. The equation connects \({\rm{X}}\) and \({\rm{A}}\) through scalar multiplication and addition.
Understanding the Given Matrix and Equation
The matrix \({\rm{A}}\) provided is:
| \({\rm{A}} = \left[ {\begin{array}{} 1 & 0 & { - 2}\\ 2 & { - 3} & 4 \end{array}} \right]\) |
The equation we must satisfy is:
\(2X + 3A = 0\)
In this equation, \(0\) represents the zero matrix, which has the same dimensions as matrices \({\rm{A}}\) and \({\rm{X}}\). Matrix \({\rm{X}}\) will also be a 2x3 matrix.
Calculating Matrix X
To find the matrix \({\rm{X}}\), we need to isolate it in the equation \(2X + 3A = 0\). Here's how we rearrange it:
- Start with the given equation: \(2X + 3A = 0\)
- Subtract \(3A\) from both sides to get \(2X\) by itself: \(2X = -3A\)
- Divide both sides by 2 (or multiply by the scalar \(\frac{1}{2}\)) to solve for \(X\): \(X = \frac{-3}{2}A\)
This calculation involves multiplying every element in matrix \({\rm{A}}\) by the scalar \(-\frac{3}{2}\).
Step-by-Step Calculation
Let's apply the scalar multiplication \(-\frac{3}{2}\) to each element of matrix \({\rm{A}}\):
\(X = -\frac{3}{2} \times \left[ {\begin{array}{} 1 & 0 & { - 2}\\ 2 & { - 3} & 4 \end{array}} \right]\)
We perform the multiplication element by element:
- First row, first column: \(-\frac{3}{2} \times 1 = -\frac{3}{2}\)
- First row, second column: \(-\frac{3}{2} \times 0 = 0\)
- First row, third column: \(-\frac{3}{2} \times (-2) = \frac{6}{2} = 3\)
- Second row, first column: \(-\frac{3}{2} \times 2 = -\frac{6}{2} = -3\)
- Second row, second column: \(-\frac{3}{2} \times (-3) = \frac{9}{2}\)
- Second row, third column: \(-\frac{3}{2} \times 4 = -\frac{12}{2} = -6\)
After performing these calculations, the resulting matrix \({\rm{X}}\) is:
| \(X = \left[ {\begin{array}{} { - \frac{3}{2}} & 0 & 3\\ { - 3} & {\frac{9}{2}} & { - 6} \end{array}} \right]\) |
Final Matrix X
The matrix \({\rm{X}}\) that satisfies the condition \(2X + 3A = 0\) is determined by performing scalar multiplication on matrix \({\rm{A}}\).
The calculated matrix \({\rm{X}}\) is:
| \(X = \left[ {\begin{array}{} { - \frac{3}{2}} & 0 & 3\\ { - 3} & {\frac{9}{2}} & { - 6} \end{array}} \right]\) |
This matrix corresponds to the fourth option provided.
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