In obtaining the solution of the system of equations $x + y + z = 7$, $x + 2y+3z = 16$ and $x + 3y+4z = 22$ by Cramer's rule, the value of $y$ is obtained by dividing $D$ by $D_2$, where $D = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 4 \end{vmatrix}$ What is the value of the determinant $D_2$?
Solving System of Equations using Cramer's Rule: Finding Determinant $D_2$
This solution explains how to find the determinant $D_2$ required for solving a given system of linear equations using Cramer's rule.
Understanding Cramer's Rule
Cramer's rule is a method for solving systems of linear equations using determinants. For a system represented as $Ax = b$, where $A$ is the coefficient matrix, $x$ is the variable vector, and $b$ is the constant vector, the solution for a specific variable (like $y$) is found using determinants.
- $D$: This is the determinant of the main coefficient matrix $A$.
- $D_y$: This is the determinant of the matrix formed by replacing the column of $y$'s coefficients in matrix $A$ with the constant vector $b$.
The value of $y$ is generally calculated as $y = \frac{D_y}{D}$.
Defining the Matrices for the Given System
The system of equations provided is:
- $x + y + z = 7$
- $x + 2y + 3z = 16$
- $x + 3y + 4z = 22$
The coefficient matrix $A$, which corresponds to the determinant $D$, is:
| $1$ | $1$ | $1$ |
| $1$ | $2$ | $3$ |
| $1$ | $3$ | $4$ |
The constant vector $b$ is:
| $7$ |
| $16$ |
| $22$ |
The question asks for the determinant $D_2$. In Cramer's rule, the determinant related to finding $y$ is denoted as $D_y$. We'll calculate this determinant, which corresponds to $D_2$ in the question's context. This matrix is formed by replacing the second column of $A$ (coefficients of $y$) with the constant vector $b$:
| $1$ | $7$ | $1$ |
| $1$ | $16$ | $3$ |
| $1$ | $22$ | $4$ |
Calculating the Determinant $D_2$ ($D_y$)
We need to compute the value of the determinant $D_2$:
$D_2 = \begin{vmatrix} 1 & 7 & 1 \\ 1 & 16 & 3 \\ 1 & 22 & 4 \end{vmatrix}$Expanding the determinant along the first row gives:
$D_2 = 1 \cdot \begin{vmatrix} 16 & 3 \\ 22 & 4 \end{vmatrix} - 7 \cdot \begin{vmatrix} 1 & 3 \\ 1 & 4 \end{vmatrix} + 1 \cdot \begin{vmatrix} 1 & 16 \\ 1 & 22 \end{vmatrix}$Calculate the determinants of the 2x2 sub-matrices:
- The determinant of the first 2x2 matrix is $(16 \times 4) - (3 \times 22) = 64 - 66 = -2$.
- The determinant of the second 2x2 matrix is $(1 \times 4) - (3 \times 1) = 4 - 3 = 1$.
- The determinant of the third 2x2 matrix is $(1 \times 22) - (16 \times 1) = 22 - 16 = 6$.
Substitute these values back into the expansion formula:
$D_2 = 1 \cdot (-2) - 7 \cdot (1) + 1 \cdot (6)$ $D_2 = -2 - 7 + 6$ $D_2 = -9 + 6$ $D_2 = -3$Final Answer
The calculated value of the determinant $D_2$ is -3.
Important Questions from Matrices and Determinants
What is the value of the determinant of the inverse of the matrix $\begin{bmatrix} -4 & -5 \\ 2 & 2 \end{bmatrix}$?
Consider the following in respect of non-singular matrices $A$ and $B$ :
I. $(AB)^{-1} = A^{-1}B^{-1}$
II. $(BA)(AB)^{-1} = I$, where $I$ is the identity matrix
III. $(AB)^T = A^T B^T$
How many of the above are correct?
Consider the following statements :
Statement-I :
If $X$ is an $n \times n$ matrix, then $\det(mX) = m^n \det(X)$, where $m$ is a scalar.
Statement-II :
If $Y$ is a matrix obtained from $X$ by multiplying any row or column by a scalar $m$, then $\det(Y) = m\det(X)$.
Which one of the following is correct in respect of the above statements?
Consider the following statements about the matrix $M = \begin{vmatrix} 71 & 23 & 48 \\ 57 & 28 & 29 \\ 65 & 17 & 48 \end{vmatrix}$
Statement-I : The inverse of $M$ does not exist.
Statement-II : $M$ is non-singular.
Which one of the following is correct in respect of the above statements?
- Let $f(x) = \begin{vmatrix} 3x^2 & \cos x & -\sin x \\ 6 & -1 & 0 \\ q & q^2 & q^3 \end{vmatrix}$ where q is any constant, then what is $\frac{d^2}{dx^2}(f(x))$ at $x = 0$ equal to ?
