If the system of equations $2x + \lambda y + 3z = 5$ $3x + 2y - z = 7$ $4x + 5y + \mu z = 9$ has infinitely many solutions, then $(\lambda^2 + \mu^2)$ is equal to:

Step 1: Formulate the augmented matrix

To solve a system of linear equations, we can represent it using an augmented matrix. This matrix combines the coefficients of the variables and the constants on the right-hand side of the equations. The vertical line separates the coefficient matrix from the constant terms.

Step 2: Condition for infinitely many solutions

For a system of linear equations to have infinitely many solutions, two conditions must be met: the determinant of the coefficient matrix ($A$) must be zero, and the determinants of the matrices formed by replacing a column of $A$ with the constant terms ($A_x, A_y, A_z$) must also be zero. This ensures that the equations are consistent and dependent.

Step 3: Calculate determinant of coefficient matrix

We calculate the determinant of the coefficient matrix and set it to zero. This will give us a relationship between $\lambda$ and $\mu$. The determinant is calculated as $2(2\mu - (-1)5) - \lambda(3\mu - (-1)4) + 3(3(5) - 2(4))$.

Step 4: Solve for the relationship between $\lambda$ and $\mu$

Expanding the determinant from the previous step, we get an algebraic equation involving $\lambda$ and $\mu$. This equation represents the condition for the determinant of the coefficient matrix to be zero.

Step 5: Calculate determinant of $A_x$

Now, we replace the first column of the coefficient matrix with the constant terms to form $A_x$. We calculate its determinant and set it to zero. This will give us another equation involving $\lambda$ and $\mu$.

Step 6: Solve for $\lambda$ and $\mu$

Expanding the determinant of $A_x$ and simplifying, we get a second equation: $7\lambda\mu + 9\lambda - 10\mu - 76 = 0$. We now have two equations with two variables, $\lambda$ and $\mu$. We can solve these simultaneously. By comparing the coefficients of $\lambda\mu$ and $\lambda$ in both equations, we can deduce that $\lambda = 2$ and $\mu = 5$.

Step 7: Calculate $\lambda^2 + \mu^2$

Finally, we substitute the values of $\lambda = 2$ and $\mu = 5$ into the expression $\lambda^2 + \mu^2$ to find the required value.

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