Question

Math

Posted 6 months ago

```
Determine the value(s) of $x$ such that
$\begin{array}{l}
{\left[\begin{array}{lll}
x & 2 & 1
\end{array}\right]\left[\begin{array}{ccc}
3 & 2 & -3 \\
2 & 0 & 3 \\
-3 & 42 & 2
\end{array}\right]\left[\begin{array}{c}
x \\
-1 \\
4
\end{array}\right]=[0]} \\
x=(-2,-1 / 6)
\end{array}$
Note: If there is more than one value separate them by commas.
```

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Answer from Sia

Posted 6 months ago

Solution by Steps

step 1

To find the value(s) of $x$, we need to multiply the given matrices and set the product equal to the zero matrix

step 2

The multiplication of the 1x3 matrix $[x \ 2 \ 1]$ with the 3x4 matrix is performed by multiplying each element of the row by the corresponding elements of the columns and summing the results

step 3

The product of the first matrix and the second matrix is a 1x4 matrix. We set each element of this resulting matrix equal to zero

step 4

The first element of the resulting matrix is $3x + 4 - 3 = 0$

step 5

The second element of the resulting matrix is $2x + 0 + 42 = 0$

step 6

The third element of the resulting matrix is $-3x + 0 + 2 = 0$

step 7

Solving the first equation $3x + 4 - 3 = 0$ gives us $x = -1/3$

step 8

Solving the second equation $2x + 42 = 0$ gives us $x = -21$

step 9

Solving the third equation $-3x + 2 = 0$ gives us $x = 2/3$

step 10

However, $x$ must satisfy all three equations simultaneously. We find that there is no single value that satisfies all three equations, which means there is an inconsistency in the system

Answer

There is no solution for $x$ as the system of equations is inconsistent.

Key Concept

Matrix multiplication and solving a system of linear equations

Explanation

The value(s) of $x$ are determined by solving the system of linear equations obtained from setting the product of the given matrices equal to the zero matrix. However, if the system is inconsistent, as in this case, there is no solution for $x$.

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