Find the value of 'x' if :
`|{:(3,x),(x,1):}|=|{:(5,2),(4,-1):}|`

Answer

To find the value of \( x \) such that

\[ \left| \begin{array}{cc} 3 & x \\ x & 1 \end{array} \right| = \left| \begin{array}{cc} 5 & 2 \\ 4 & -1 \end{array} \right| \]

we will calculate the determinants of both matrices and set them equal to each other.

### Step 1: Calculate the determinant of the left-hand side (LHS)

The determinant of a \( 2 \times 2 \) matrix

\[ \left| \begin{array}{cc} a & b \\ c & d \end{array} \right| = ad - bc \]

For the LHS, we have:

\[ \left| \begin{array}{cc} 3 & x \\ x & 1 \end{array} \right| = (3)(1) - (x)(x) = 3 - x^2 \]

### Step 2: Calculate the determinant of the right-hand side (RHS)

For the RHS, we have:

\[ \left| \begin{array}{cc} 5 & 2 \\ 4 & -1 \end{array} \right| = (5)(-1) - (2)(4) = -5 - 8 = -13 \]

### Step 3: Set the determinants equal to each other

Now we set the LHS equal to the RHS:

\[ 3 - x^2 = -13 \]

### Step 4: Solve for \( x^2 \)

Rearranging the equation gives:

\[ 3 + 13 = x^2 \]

\[ x^2 = 16 \]

### Step 5: Take the square root to find \( x \)

Taking the square root of both sides, we find:

\[ x = \pm 4 \]

Thus, the values of \( x \) are \( 4 \) and \( -4 \).

### Final Answer:

The values of \( x \) are \( 4 \) and \( -4 \).

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