Determinant of a Matrix with Trigonometric Functions: Solve for x

Published on June 10, 2025

Linear-Algebra

Solve for x in a determinant problem involving trigonometric functions. The solution involves calculating the determinant of a 3x3 matrix, using trigonometric identities, and solving a trigonometric equation.

Question

Resuélvelo

Answer

The user wants to solve a math problem involving determinants.
The problem asks to find the value of

such that

, where

is the matrix:

First, let’s calculate the determinant of matrix

. We can use the Sarrus’ rule or cofactor expansion. Let’s use cofactor expansion along the third column because it contains ones, which might simplify calculations.

where

is the cofactor of the element

Using the identity

, we get

Now, sum the cofactors:

Using the double angle identity

, we have:

The problem states that

.
So, we need to solve the equation:

To find

, we take the inverse sine (arcsin) of 0.91:

Using a calculator,

radians.
So,

radians.

Therefore,

radians.

Since the sine function is periodic, there are infinitely many solutions. The general solution for

, where

is an integer.
So,

.
For

, so

.
For

, so

.
The problem asks for “el valor del escalar ‘x’”, implying a single principal value. Usually, this refers to the principal value from the arcsin function.

Let’s double check the determinant calculation.
Using Sarrus’ rule: