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Mathematics · Class 12 Science

Ch 4Determinants — Class 12 Mathematics, concept-first.

Associated with every square matrix is a unique number called its determinant, written or . Unlike a matrix (an array of numbers), a determinant is a single scalar value computed from the entries of the matrix according to a fixed rule.

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Key concepts

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Adjoint and Inverse of a Matrix

For a square matrix of order , the cofactor of is the signed minor (the minor is the determinant left after deleting row and column ).

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

1

Determinant of a Square Matrix (2×2 and 3×3)

Associated with every square matrix is a unique number called its determinant, written or . Unlike a matrix (an array of numbers), a determinant is a single scalar value computed from the entries of t…

2

Properties of Determinants

Expanding a determinant directly, entry by entry, becomes tedious once the entries are large or algebraic.

3

Minors and Cofactors

For a square matrix of order , the minor of the element is the determinant of the matrix left after deleting the row and the column that sits in -- that is, row and column .

4

Area of a Triangle Using Determinants

Given a triangle with vertices , elementary coordinate geometry (splitting the triangle into trapezoids against the -axis, or using the shoelace method) gives its area as This same expression can be w…

5

Adjoint of a Square Matrix

For a square matrix , first form the matrix of cofactors of -- the matrix obtained by replacing every entry with its cofactor (Section 3).

6

Inverse of a Square Matrix (via Adjoint)

A square matrix is called non-singular if , and singular if . Only a non-singular square matrix has an inverse: the inverse of , written , is the unique matrix satisfying If is singular, no such matri…

7

Consistency of a System of Linear Equations

A system of linear equations such as can be written compactly as a single matrix equation , where is called the coefficient matrix.

8

Solving Linear Systems Using the Matrix Method

Given a system with the coefficient matrix, the column of unknowns and the column of constants, provided (Section 7 confirms the system is then consistent with a unique solution), the solution is foun…

Summary

This chapter built the theory of determinants for square matrices up to order and used them to solve linear systems.

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 17 questions17 questions
  1. Q1If det([[2, 3], [4, 5]]) = det([[x, 3], [2x, 5]]), find the value of x.Preview
  2. Q2Show that det([[1+a, 1, 1], [1, 1+b, 1], [1, 1, 1+c]]) = abc (1 + 1/a + 1/b + 1/c), (abc not equal to 0). **OR** Using Cramer's rule solve t…Preview
  3. Q3If two rows or two columns of a determinant are identical then value of the determinant is (a) 0 (b) 2 (c) -1 (d) 1Preview
  4. Q4If |5 4; 3 2| = |2x 7; x 3| (determinants of the two 2×2 matrices are equal), find the value of x.Preview
  5. Q5Show that |1+a 1 1; 1 1+b 1; 1 1 1+c| (3×3 determinant) = abc(1 + 1/a + 1/b + 1/c). **OR** Show that |1 x x²; x² 1 x; x x² 1| (3×3 determina…Preview
  6. Q6Show that |0 a b; -a 0 c; -b -c 0| = 0.Preview
  7. Q7Show that |a a+b a+b+c; 2a 3a+2b 4a+3b+2c; 3a 6a+3b 10a+6b+3c| = a³. **OR** a, b, c are real numbers and |b+c c+a a+b; c+a a+b b+c; a+b b+c…Preview
  8. Q8A is a square matrix of order 3. The value of |kA| is equal to (k is a constant) (a) k|A| (b) k^2|A| (c) k^3|A| (d) 3k|A|Preview
  9. Q9If determinant | -5 5 10; 5 -5 x; 0 10 5 | = 0, find x.Preview
  10. Q10Prove that determinant | 2a a-b-c 2a ; 2b 2b b-c-a ; c-a-b 2c 2c | = (a+b+c)³. **OR** If a ≠ p, b ≠ q, c ≠ r and determinant | p b c ; a q c…Preview
  11. Q11If 1, ω and ω² are the cube roots of unity, then find the value of k for which the matrix (1 ω k; ω k 1; k 1 ω) is singular.Preview
  12. Q12Prove that one root of the equation |x+a b c; b x+c a; c a x+b| = 0 is -(a+b+c).Preview
  13. Q13Using properties of determinants, prove that |a+b+c -c -b; -c a+b+c -a; -b -a a+b+c| = 2(a+b)(b+c)(c+a). **OR** Solve by Cramer's rule: 1/x+…Preview
  14. Q14If $\begin{vmatrix} -x^2 & xy & xz \\ xy & -y^2 & yz \\ xz & yz & -z^2 \end{vmatrix} = \lambda\, x^2 y^2 z^2$, then the value of $\lambda$ i…Preview
  15. Q15The system of equations $kx + y + z = 1$, $x + ky + z = k$ and $x + y + kz = k^2$ will have unique solution when (a) $k \neq 1$ (b) $k \neq…Preview
  16. Q16If $A$ is a square matrix of order 3 and $|A| = 7$, then the value of $|2A^T|$ is (a) $32$ (b) $28$ (c) $16$ (d) $56$Preview
  17. Q17If the inverse of a matrix $A$ of order $3 \times 3$ exists and $|A| = 5$, then the value of $|adj\, A|$ is (a) $20$ (b) $15$ (c) $5$ (d) $2…Preview

More questions

32 Q
+Show 2 questions2 questions
  1. Q31If $A=\begin{pmatrix}1&2\\2&3\end{pmatrix}$, verify that $A(\operatorname{adj}A)=|A|I=(\operatorname{adj}A)A$.Free
  2. Q32Using properties of determinants, prove that $\begin{vmatrix}1&x&x^2\\1&y&y^2\\1&z&z^2\end{vmatrix}=(x-y)(y-z)(z-x)$.Preview
+Show 8 questions8 questions
  1. Example 1Evaluate the determinant $\begin{vmatrix}3&-1\\2&4\end{vmatrix}$.Free
  2. Example 2Evaluate $\begin{vmatrix}2&3&1\\0&-1&4\\5&2&-3\end{vmatrix}$ by expanding along the first row.Free
  3. Example 3Using properties of determinants, evaluate $\begin{vmatrix}101&102&103\\104&105&106\\107&108&109\end{vmatrix}$ without direct expansion.Free
  4. Example 4Write the minors and cofactors of the elements of the first row of $\begin{vmatrix}2&-3&5\\6&0&4\\1&5&-7\end{vmatrix}$.Preview
  5. Example 5Using determinants, find the area of the triangle whose vertices are $(3,8)$, $(-4,2)$ and $(5,-1)$.Preview
  6. Example 6Find the adjoint of $A=\begin{pmatrix}1&2\\3&4\end{pmatrix}$ and verify that $A(\operatorname{adj}A)=|A|I$.Preview
  7. Example 7Using the adjoint method, find the inverse of $A=\begin{pmatrix}2&1&1\\1&2&1\\1&1&2\end{pmatrix}$.Preview
  8. Example 8Using the matrix $A$ and $A^{-1}$ found in Example 7, solve the system $2x+y+z=7,\ x+2y+z=8,\ x+y+2z=9$ by the matrix method.Preview
+Show 4 questions4 questions
  1. Q9Evaluate $\begin{vmatrix}5&-2\\3&4\end{vmatrix}$.Free
  2. Q10Evaluate $\begin{vmatrix}0&1\\-1&0\end{vmatrix}$.Free
  3. Q11Evaluate $\begin{vmatrix}1&0&2\\-1&3&1\\2&1&0\end{vmatrix}$ by expanding along the first column.Preview
  4. Q12Evaluate $\begin{vmatrix}2&3&-1\\4&-1&2\\3&2&1\end{vmatrix}$.Preview
+Show 4 questions4 questions
  1. Q13Without expanding, evaluate $\begin{vmatrix}3&4&5\\3&4&5\\1&2&3\end{vmatrix}$, stating the property used.Free
  2. Q14Evaluate $\begin{vmatrix}6&3&9\\2&4&8\\1&5&7\end{vmatrix}$ by first taking the common factor $3$ out of row 1.Free
  3. Q15Without full expansion, show that $\begin{vmatrix}1&2&3\\4&5&6\\7&8&9\end{vmatrix}=0$ using a suitable row operation.Preview
  4. Q16Verify the property that interchanging two rows reverses the sign of a determinant, using $\begin{vmatrix}2&5\\1&3\end{vmatrix}$ and $\begin…Preview
+Show 3 questions3 questions
  1. Q17Find the minors and cofactors of every element of $\begin{pmatrix}5&20\\0&-1\end{pmatrix}$.Free
  2. Q18Find the minors and cofactors of the elements $a_{21}, a_{22}, a_{23}$ of $\begin{vmatrix}3&-1&2\\4&0&-3\\1&5&2\end{vmatrix}$.Preview
  3. Q19Find the cofactor of the element $a_{32}$ in the matrix $\begin{pmatrix}2&-3&5\\6&0&4\\1&5&-7\end{pmatrix}$.Preview
+Show 3 questions3 questions
  1. Q20Find the area of the triangle with vertices $(1,0),(6,0),(4,3)$ using determinants.Free
  2. Q21Find the value(s) of $k$ if the area of the triangle with vertices $(k,0),(4,0),(0,2)$ is $4$ square units.Preview
  3. Q22Using determinants, show that the points $(a,b+c),(b,c+a),(c,a+b)$ are collinear.Preview
+Show 4 questions4 questions
  1. Q23Find the adjoint of $A=\begin{pmatrix}3&-2\\1&4\end{pmatrix}$ and verify $A(\operatorname{adj}A)=|A|I$.Free
  2. Q24Find $A^{-1}$ for $A=\begin{pmatrix}2&-1\\-3&4\end{pmatrix}$ using the adjoint method, and verify $AA^{-1}=I$.Free
  3. Q25Show that $A=\begin{pmatrix}2&3\\1&2\end{pmatrix}$ is non-singular and find $A^{-1}$.Preview
  4. Q26Using the adjoint method, find the inverse of $A=\begin{pmatrix}1&0&0\\3&3&0\\5&2&-1\end{pmatrix}$.Preview
+Show 4 questions4 questions
  1. Q27Solve the system $2x+3y=7,\ x-y=1$ using the matrix method.Free
  2. Q28Solve the system $5x+2y=4,\ 7x+3y=5$ using the matrix method.Free
  3. Q29Solve the system $x+y+z=6,\ 2x+y-z=1,\ x-y+2z=5$ using the matrix method.Preview
  4. Q30Using determinants, examine whether the system $x-2y=4,\ 2x-4y=5$ is consistent. If it is not, explain why.Preview