SATHEE: Exercise 04

04 Determinants

Exercise 04

Question:

Write minors and cofactors of the elements of following determinants (i) $| \begin{matrix} 2 & -4 \\ 0 & 3 \end{matrix} |$ (ii) $| \begin{matrix} a & c \\ b & d \end{matrix} |$

Answer:

Answer: (i) Minors: M11 = 3, M12 = -4, M21 = 0, M22 = 2 Cofactors: C11 = -3, C12 = 4, C21 = 0, C22 = -2

(ii) Minors: M11 = d, M12 = -c, M21 = -b, M22 = a Cofactors: C11 = -d, C12 = c, C21 = b, C22 = -a

Question:

Find the Minors and Cofactors of the elements of the following determinants: (i) $| \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} |$ (ii) $| \begin{matrix} 1 & 0 & 4 \\ 3 & 5 & -2 \\ 0 & 1 & 2 \end{matrix} |$

Answer:

(i) Minor of 1 = 1; Cofactor of 1 = 1 Minor of 0 = 0; Cofactor of 0 = 0 Minor of 0 = 0; Cofactor of 0 = 0

(ii) Minor of 1 = -20; Cofactor of 1 = -20 Minor of 0 = 0; Cofactor of 0 = 0 Minor of 4 = -8; Cofactor of 4 = 8 Minor of 3 = 10; Cofactor of 3 = -10 Minor of 5 = -2; Cofactor of 5 = 2 Minor of -2 = 10; Cofactor of -2 = -10 Minor of 0 = 0; Cofactor of 0 = 0 Minor of 1 = -20; Cofactor of 1 = 20 Minor of 2 = 8; Cofactor of 2 = -8

Question:

Using cofactors of elements of second row, evaluate Δ= $| \begin{matrix} 5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3 \end{matrix} |$

Answer:

Step 1: Calculate the cofactors of the elements of the second row.

Cofactor of 2 = (−3)

Cofactor of 0 = 2

Cofactor of 1 = (−1)

Step 2: Multiply the cofactors with the elements of the second row and add them up.

(−3) × 2 + 2 × 0 + (−1) × 1 = (−3) + 0 + (−1) = −4

Step 3: Calculate the determinant by multiplying the sum from Step 2 with the determinant of the original matrix.

Δ = (−4) × $| \begin{matrix} 5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3 \end{matrix} |$

Δ = (−4) × (−14) = 56

Question:

If Δ= $| \begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} |$ and Aij is cofactors of aij, then the value of Δ is given by A a11A31+a12A32+a13A33 B a11A11+a12A21+a13A31 C a21A11+a22A12+a23A13 D a11A11+a21A21+a31A31

Answer:

A. A11A31 + a12A32 + a13A33 B. a11A11 + a12A21 + a13A31 C. a21A11 + a22A12 + a23A13 D. a11A11 + a21A21 + a31A31

Question:

Using cofactors of elements of third column evaluate Δ= $| \begin{matrix} 1 & x & yz \\ 1 & y & zx \\ 1 & z & xy \end{matrix} |$

Answer:

Step 1: Find the cofactors of elements of third column.

C11 = yz, C12 = -zx, C13 = xy

Step 2: Evaluate Δ using the cofactors.

Δ = |1 x yz| |1 y -zx| |1 z xy|

Δ = yz(-zx) + (-zx)xy + xy(yz)

Δ = -yzxz + xyz2

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JEE NCERT Solutions (Mathematics)

01 Relations and Functions

02 Inverse Trigonometric Functions

03 Matrices

04 Determinants

05 Continuity and Differentiability

06 Application of Derivatives

07 Integrals

08 Application of Integrals

09 Vectors

10 Three Dimensional Geometry

11 Linear Programming

12 Probability

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