For a system of three linear equations \(AX = B\) in x, y, z, let \(\Delta = |A|\), and let \(\Delta_x\), \(\Delta_y\), \(\Delta_z\) be the determinants used in Cramer’s rule. Which of the following statements are correct?
(A) If \(\Delta \ne 0\), the system has the unique solution \(x = \displaystyle\frac{\Delta_x}{\Delta}\), \(y = \displaystyle\frac{\Delta_y}{\Delta}\), \(z = \displaystyle\frac{\Delta_z}{\Delta}\).
(B) If \(\Delta = 0\) and at least one of \(\Delta_x\), \(\Delta_y\), \(\Delta_z\) is non-zero, the system has no solution.
(C) \(\Delta_x\) is obtained by replacing the first row of A by the elements of B.
(D) If \(\Delta \ne 0\), the system can have infinitely many solutions.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (B) only
In Cramer’s rule, \(\Delta_x\), \(\Delta_y\), \(\Delta_z\) are formed by replacing the x-column, y-column and z-column of A (its 1st, 2nd and 3rd columns) by the column of constants B. The rule rests on the three relations
\(\Delta \cdot x = \Delta_x,\quad \Delta \cdot y = \Delta_y,\quad \Delta \cdot z = \Delta_z\)
which every solution of the system satisfies.
So \(\Delta \ne 0\) gives exactly one solution, and \(\Delta = 0\) with some \(\Delta_x\), \(\Delta_y\) or \(\Delta_z\) non-zero gives no solution; the replacement is made in a column, not a row.
Checking the statements
(A) If \(\Delta \ne 0\), the system has the unique solution \(x = \displaystyle\frac{\Delta_x}{\Delta}\), \(y = \displaystyle\frac{\Delta_y}{\Delta}\), \(z = \displaystyle\frac{\Delta_z}{\Delta}\). — Correct. Dividing the three relations by \(\Delta \ne 0\) gives these values.
(B) If \(\Delta = 0\) and at least one of \(\Delta_x\), \(\Delta_y\), \(\Delta_z\) is non-zero, the system has no solution. — Correct. A relation such as \(0 \cdot x = \Delta_x \ne 0\) can never hold, so the system is inconsistent.
(C) \(\Delta_x\) is obtained by replacing the first row of A by the elements of B. — Incorrect. The first column (the x-column) is replaced, not the first row.
(D) If \(\Delta \ne 0\), the system can have infinitely many solutions. — Incorrect. A non-zero \(\Delta\) always gives exactly one solution.