Determinants — JEE Mains Mathematics

Formulas (2)

• The third equation as a combination · Dependent third equation
  $$E_3=pE_1+qE_2$$
• When the determinant vanishes for every value · Dependent rows
  $$R_3=pR_1+qR_2\ \Rightarrow\ \text{consistent exactly when } b_3=pb_1+qb_2$$

Watch out for (2)

• Match the right-hand side too
  → The third equation as a combination
• Look for dependence before expanding
  → When the determinant vanishes for every value

Formulas (2)

• The coefficient determinant and one Cramer determinant · Cramer's rule
  $$x=\frac{\Delta_x}{\Delta},\quad y=\frac{\Delta_y}{\Delta},\quad z=\frac{\Delta_z}{\Delta}$$
• Elimination instead of determinants · The last equation decides
  $$kz=c:\quad k\neq0\ \text{unique};\ k=c=0\ \text{infinite};\ k=0\neq c\ \text{none}$$

Watch out for (2)

• Confirm consistency
  → The coefficient determinant and one Cramer determinant
• Keep the right-hand sides
  → Elimination instead of determinants

Formulas (2)

• The value that gives no solution · Inconsistency
  $$\Delta=0,\quad(\Delta_x,\Delta_y,\Delta_z)\neq(0,0,0)$$
• Counting the values · Count what survives
  $$\#\{\theta:\Delta(\theta)=0,\ \text{inconsistent}\}$$

Watch out for (2)

• A root of the determinant may give infinitely many
  → The value that gives no solution
• Candidates are not answers
  → Counting the values

Formulas (2)

• The full classification · Three cases
  $$\Delta\neq0\Rightarrow\text{unique};\quad\Delta=0\Rightarrow\text{infinite or none}$$
• The unique solution · Cramer's rule
  $$x=\frac{\Delta_x}{\Delta},\ y=\frac{\Delta_y}{\Delta},\ z=\frac{\Delta_z}{\Delta}\quad(\Delta\neq0)$$

Watch out for (2)

• Two options can be wrong
  → The full classification
• Count ordered choices
  → The unique solution

Formulas (2)

• An angle in the coefficients · Non-trivial solutions
  $$A\mathbf x=\mathbf0\ \text{has }\mathbf x\neq\mathbf0\ \text{exactly when}\ |A|=0$$
• An algebraic constant · Symmetric determinant
  $$\begin{vmatrix}k&1&1\\1&k&1\\1&1&k\end{vmatrix}=(k+2)(k-1)^2$$

Watch out for (2)

• Open or closed interval
  → An angle in the coefficients
• Every root counts
  → An algebraic constant

Formulas (2)

• Scalar multiples and adjoints · Adjoint rules
  $$|kA|=k^n|A|,\qquad|\mathrm{adj}A|=|A|^{n-1}$$
• Products, inverses and cofactors · Product rule
  $$|AB|=|A|\,|B|$$

Watch out for (2)

• The power depends on the order
  → Scalar multiples and adjoints
• Divide by a negative determinant
  → Products, inverses and cofactors

Formulas (2)

• Factor out and create zeros · Row operation
  $$R_i\to R_i-kR_j:\ \Delta\ \text{unchanged}$$
• Expand and compare · Vandermonde
  $$\begin{vmatrix}1&a&a^2\\1&b&b^2\\1&c&c^2\end{vmatrix}=(a-b)(b-c)(c-a)$$

Watch out for (2)

• Scaling a row scales the determinant
  → Factor out and create zeros
• Sign of the cofactor
  → Expand and compare

Formulas (2)

• Trigonometric determinants · Range of a sinusoid
  $$a\sin\theta+b\cos\theta\in\left[-\sqrt{a^2+b^2},\ \sqrt{a^2+b^2}\right]$$
• Derivatives, integrals and limits · Derivative of a determinant
  $$\frac{d}{dx}\begin{vmatrix}f_1&f_2\\g_1&g_2\end{vmatrix}=\begin{vmatrix}f_1'&f_2'\\g_1&g_2\end{vmatrix}+\begin{vmatrix}f_1&f_2\\g_1'&g_2'\end{vmatrix}$$

Watch out for (2)

• Simplify before expanding
  → Trigonometric determinants
• Differentiate one row at a time
  → Derivatives, integrals and limits