Answer :
To solve the given equation \(\left|\begin{array}{cc}2 x & 4 \\ -1 & x\end{array}\right|=\left|\begin{array}{rr}6 & -3 \\ 2 & 1\end{array}\right|\) for \(x\), we need to determine the determinants of the matrices on both sides of the equation and then solve the resulting equation.
### Step 1: Calculate the determinant of the left matrix
We'll start by calculating the determinant of the matrix on the left:
[tex]\[ \left|\begin{array}{cc}2 x & 4 \\ -1 & x\end{array}\right| = (2x)(x) - (-1)(4) = 2x^2 + 4 \][/tex]
### Step 2: Calculate the determinant of the right matrix
Next, we calculate the determinant of the matrix on the right:
[tex]\[ \left|\begin{array}{rr}6 & -3 \\ 2 & 1\end{array}\right| = (6)(1) - (-3)(2) = 6 + 6 = 12 \][/tex]
### Step 3: Set the determinants equal
Now we set the determinants equal to each other:
[tex]\[ 2x^2 + 4 = 12 \][/tex]
### Step 4: Solve the equation for \(x\)
To find \(x\), we need to solve the equation:
[tex]\[ 2x^2 + 4 = 12 \][/tex]
Subtract 4 from both sides to simplify:
[tex]\[ 2x^2 = 8 \][/tex]
Divide both sides by 2 to isolate \(x^2\):
[tex]\[ x^2 = 4 \][/tex]
Take the square root of both sides:
[tex]\[ x = \pm 2 \][/tex]
### Conclusion
The solutions to the equation are:
[tex]\[ x = -2 \quad \text{and} \quad x = 2 \][/tex]
Thus, the values of [tex]\(x\)[/tex] that satisfy the equation are [tex]\(\boxed{-2 \text{ and } 2}\)[/tex].
### Step 1: Calculate the determinant of the left matrix
We'll start by calculating the determinant of the matrix on the left:
[tex]\[ \left|\begin{array}{cc}2 x & 4 \\ -1 & x\end{array}\right| = (2x)(x) - (-1)(4) = 2x^2 + 4 \][/tex]
### Step 2: Calculate the determinant of the right matrix
Next, we calculate the determinant of the matrix on the right:
[tex]\[ \left|\begin{array}{rr}6 & -3 \\ 2 & 1\end{array}\right| = (6)(1) - (-3)(2) = 6 + 6 = 12 \][/tex]
### Step 3: Set the determinants equal
Now we set the determinants equal to each other:
[tex]\[ 2x^2 + 4 = 12 \][/tex]
### Step 4: Solve the equation for \(x\)
To find \(x\), we need to solve the equation:
[tex]\[ 2x^2 + 4 = 12 \][/tex]
Subtract 4 from both sides to simplify:
[tex]\[ 2x^2 = 8 \][/tex]
Divide both sides by 2 to isolate \(x^2\):
[tex]\[ x^2 = 4 \][/tex]
Take the square root of both sides:
[tex]\[ x = \pm 2 \][/tex]
### Conclusion
The solutions to the equation are:
[tex]\[ x = -2 \quad \text{and} \quad x = 2 \][/tex]
Thus, the values of [tex]\(x\)[/tex] that satisfy the equation are [tex]\(\boxed{-2 \text{ and } 2}\)[/tex].
