Class 12 NCERT Solutions - Mathematics Part I - Chapter 4 Determinants - Exercise 4.4

Exercise 4.4 of Chapter 4 in the NCERT Class 12 Mathematics Part I textbook focuses on advanced applications of determinants and their properties. This exercise builds upon the foundational knowledge established in previous sections, challenging students to apply determinant techniques to more complex problems.

Students will encounter problems requiring a deep understanding of determinant properties, matrix operations, and geometric interpretations. This exercise aims to enhance students' analytical skills and prepare them for higher-level mathematics by presenting determinants as a powerful tool for solving sophisticated mathematical problems.

What are Determinants?

A determinant is a scalar value that is computed from the elements of a square matrix. Determinants have important applications in linear algebra, including solving systems of linear equations, finding the inverse of a matrix, and understanding the properties of linear transformations. The determinant is denoted as det(A) or |A|, where A is the matrix.

Write minors and cofactors of the elements of the following determinants: 

Question 1. 

(i) \begin{vmatrix}3&-4\\0&3\end{vmatrix}  

(ii) \begin{vmatrix}a&c\\b&d\end{vmatrix}      

Solution: 

(i) \begin{vmatrix}3&-4\\0&3\end{vmatrix}  

Finding minors of the elements of the determinant: 

Let us assume M_ij is Minors of elements a_ij

M₁₁ = Minor of elements a₁₁ = 3

M₁₂ = Minor of elements a₁₂ = 0

M₂₁ = Minor of elements a₂₁ = −4

M₂₂ = Minor of elements a₂₂ = 2

Finding cofactor of a_ij

Let us assume cofactor of a_ij is A_ij M_ij

A₁₁ = (−1)¹⁺¹ M₁₁ = (−1)² (3) = 3

A₁₂ = (−1)¹⁺² M₁₂ = (−1)³ (0) = 0

A₂₁ = (−1)²⁺¹ M₂₁ = (−1)³ (−4) = 4

A₂₂ = (−1)²⁺² M₂₂ = (−1)⁴ (2) = 2

(ii) \begin{vmatrix}a&c\\b&d\end{vmatrix}  

Finding minors of the elements of the determinant:

Let us assume M_ij is Minors of elements a_ij

M₁₁ = Minor of element a₁₁ = d

M₁₂ = Minor of elements a₁₂ = b

M₂₁ = Minor of elements a₂₁ = c

M₂₂ = Minor of elements a₂₂ = a

Finding cofactor of a_ij

Let us assume cofactor of a_ij is A_ij, which is (−1)^i+j M_ij

A₁₁ = (−1)¹⁺¹ M₁₁ = (−1)² (d) = d

A₁₂ = (−1)¹⁺² M₁₂ = (−1)³ (b) = −b

A₂₁ = (−1)²⁺¹ M₂₁ = (−1)³ (c) = −c

A₂₂ = (−1)²⁺² M₂₂ = (−1)⁴ (a) = a

Question 2.

(i)\begin{vmatrix}1&0&0\\0&1&0\\0&0&1\end{vmatrix}

(ii)\begin{vmatrix}1&0&4\\3&5&-5\\0&2&2\end{vmatrix}

Solution:

(i) \begin{vmatrix}1&0&0\\0&1&0\\0&0&1\end{vmatrix}

Let us find the Minors and cofactors of the elements:

Assume, M_ij is minor of element a_ij and A_ij is cofactor of a_ij

M₁₁ = Minor of elements a₁₁ =\begin{vmatrix} 1 & 0\\0 & 1\end{vmatrix} = 1 − 0 = 1 and A₁₁ = 1

M₁₂ = Minor of elements a₁₂ =\begin{vmatrix} 0 & 0\\0 & 1\end{vmatrix} = 0 − 0 = 0 and A₁₂ = 0

M₁₃ = Minor of elements a₁₃ =\begin{vmatrix} 0 & 1\\0 & 0\end{vmatrix} = 0 − 0 = 0 and A₁₃ = 0

M₂₁ = Minor of elements a₂₁ =\begin{vmatrix} 0 & 0\\0 & 1\end{vmatrix} = 0 − 0 = 0 and A₂₁ = 0

M₂₂ = Minor of elements a₂₂ =\begin{vmatrix} 1 & 0\\0 & 1\end{vmatrix} = 1 − 0 = 1 and A₂₂ = 1

M₂₃ = Minor of elements a₂₃ =\begin{vmatrix} 1 & 0\\0 & 0\end{vmatrix} = 0 − 0 = 0 and A₂₃ = 0

M₃₁ = Minor of elements a₃₁ =\begin{vmatrix} 0 & 0\\1 & 0\end{vmatrix} = 0 − 0 = 0 and A₃₁ = 0

M₃₂ = Minor of elements a₃₂ =\begin{vmatrix} 1 & 0\\0 & 0\end{vmatrix} = 0 − 0 = 0 and A₃₂ = 0

M₃₃ = Minor of elements a₃₃ =\begin{vmatrix} 1 & 0\\0 & 1\end{vmatrix} = 1 − 0 = 1 and A₃₃ = 1

(ii) \begin{vmatrix}1&0&4\\3&5&-5\\0&2&2\end{vmatrix}

Let us find the Minors and cofactors of the elements:

Assume, M_ij is minor of element a_ij and A_ij is cofactor of a_ij

M₁₁ = Minor of elements a₁₁ =\begin{vmatrix} 5 & -1\\1 & 2\end{vmatrix} = 10 − (−1) = 11 and A₁₁ = 11

M₁₂ = Minor of elements a₁₂ =\begin{vmatrix} 3 & -1\\0 & 2\end{vmatrix} = 6 − 0 = 6 and A₁₂ = −6

M₁₃ = Minor of elements a₁₃ =\begin{vmatrix} 3 & 5\\0 & 1\end{vmatrix} = 3 − 0 = 3 and A₁₃ = 3

M₂₁ = Minor of elements a₂₁ =\begin{vmatrix} 0 & 4\\1 & 2\end{vmatrix} = 0 − 4 = −4 and A₂₁ = 4

M₂₂ = Minor of elements a₂₂ =\begin{vmatrix} 1 & 4\\0 & 2\end{vmatrix} = 2 − 0 = 2 and A₂₂ = 2

M₂₃ = Minor of elements a₂₃ =\begin{vmatrix} 1 & 0\\0 & 1\end{vmatrix} = 1 − 0 = 1 and A₂₃ = −1

M₃₁ = Minor of elements a₃₁ =\begin{vmatrix} 0 & 4\\5 & -1\end{vmatrix} = 0 − 20 = −20 and A₃₁ = −20

M₃₂ = Minor of elements a₃₂ =\begin{vmatrix} 1 & 4\\3 & -1\end{vmatrix} = −1 − 12 = −13 and A₃₂ = 13

M₃₃ = Minor of elements a₃₃ =\begin{vmatrix} 1 & 0\\3 & 5\end{vmatrix} = 5 − 0 = 5 and A₃₃ = 5

Question 3. Using Cofactors of elements of second row, evaluate △?

\bigtriangleup=\begin{vmatrix}5&3&8\\2&0&1\\1&2&3\end{vmatrix}

Solution:

Finding the Cofactors of elements of second row:

A₂₁ = Cofactor of elements a₂₁ = (−1)²⁺¹  \begin{vmatrix} 3 & 8\\2 & 3\end{vmatrix} = (−1)³ (9 − 16) = 7

A₂₂ = Cofactor of elements a₂₂ = (−1)²⁺²  \begin{vmatrix} 5 & 8\\1 & 3\end{vmatrix} = (−1)⁴ (15 − 8) = 7

A₂₃ = Cofactor of elements a₂₃ = (−1)²⁺³  \begin{vmatrix} 5 & 3\\1 & 2\end{vmatrix} = (−1)⁵ (10 − 3) = 7

Now, △ = a₂₁ A₂₁ + a₂₂ A₂₂ + a₂₃ A₂₃ = 14 + 0 − 7 = 7

Question 4. Using Cofactors of elements of third column, evaluate △?

\bigtriangleup = \begin{vmatrix}1&x&yz\\1&y&zx\\1&z&xy\end{vmatrix}

Solution:

Finding the Cofactors of elements of third column:

A₁₃ = Cofactor of elements a₁₃ = (−1)¹⁺³  \begin{vmatrix} 1 & y\\1 & z\end{vmatrix} = (−1)⁴ (z − y) = −y

A₂₃ = Cofactor of elements a₂₃ = (−1)²⁺³  \begin{vmatrix} 1 & x\\1 & z\end{vmatrix} = (−1)⁵ (z − x) = x − z

A₃₃ = Cofactor of elements a₃₃ = (−1)³⁺³  \begin{vmatrix} 1 & x\\1 & y\end{vmatrix} = (−1)⁶ (y − x) = y − x

Now, △ = a₁₃ A₁₃ + a₂₃ A₂₃ + a₃₃ A₃₃

= yz (z − y) + zx (x − z) + xy (y − x)

= (yz² − y²z) + (xy² − xz²) + (xz² − x²y)

= (y − z)[−yz + x(y + z) − x²]

= (y − z)[−yz + x (z − x) + x (z − x)]

= (x − y)(y − x)(z − x)

Question 5. If  △=\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{vmatrix} and A_ij is cofactor of a_ij then value of △ is given by:

(A) a₁₁A₃₁ + a₁₂A₃₂ + a₁₃A₃₃

(B) a₁₁A₁₁ + a₁₂A₂₁ + a₁₃A₃₁

(C) a₂₁A₁₁ + a₂₂A₁₂ + a₂₃A₁₃

(D) a₁₁A₁₁ + a₂₁A₂₁ + a₃₁A₃₁

Solution: 

Option (D) is correct.

Summary

Exercise 4.4 of Chapter 4 on Determinants presents a comprehensive set of advanced problems that challenge students to apply their knowledge of determinants in diverse mathematical contexts. This exercise emphasizes the interconnections between determinants, matrices, geometry, and algebraic systems. Students are required to prove complex identities, solve non-linear systems of equations, and explore the geometric implications of determinant properties. The problems in this set are designed to deepen students' understanding of determinants as a versatile mathematical tool, applicable in various areas of mathematics. By working through these questions, students develop advanced problem-solving skills and gain a more profound appreciation for the elegance and power of determinants in higher mathematics.