SATHEE: Exercise 04

04 सारणिक

अभ्यास 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

अध्ययन सामग्री पर वापस

जेईई अध्ययन सामग्री (गणित)

01 संबंध एवं फलन

02 व्युत्क्रम त्रिकोणमितीय फलन

03 आव्यूह

04 सारणिक

05 सांत्यता और अवकलनीयता

06 अवकलज का अनुप्रयोग

07 समाकलन

08 समाकलन का अनुप्रयोग

09 वैक्टर

10 त्रिविमीय ज्यामिति का परिचय

11 रैखिक प्रोग्रामिंग

12 प्रायिकता

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