SATHEE: Exercise 04

04 ନିର୍ଣ୍ଣୟକାରୀ

ବ୍ୟାୟାମ 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

ଅଧ୍ୟୟନ ସାମଗ୍ରୀକୁ ଫେରନ୍ତୁ

JEE ଅଧ୍ୟୟନ ସାମଗ୍ରୀ (ଗଣିତ)

01 ସମ୍ପର୍କ ଏବଂ କାର୍ଯ୍ୟ

02 ଓଲଟା ଟ୍ରାଇଗୋନେଟ୍ରିକ୍ କାର୍ଯ୍ୟଗୁଡ଼ିକ

03 ମ୍ୟାଟ୍ରିକ୍ସ

04 ନିର୍ଣ୍ଣୟକାରୀ

05 ନିରନ୍ତରତା ଏବଂ ଭିନ୍ନତା

06 ଡେରିଭେଟିକ୍ସର ପ୍ରୟୋଗ

07 ଇଣ୍ଟିଗ୍ରାଲ୍

08 ଇଣ୍ଟିଗ୍ରାଲ୍ସର ପ୍ରୟୋଗ

09 ଭେକ୍ଟର୍

10 ତିନୋଟି ଡାଇମେନ୍ସନାଲ୍ ଜ୍ୟାମିତି

11 ରେଖା ପ୍ରୋଗ୍ରାମିଂ

12 ସମ୍ଭାବନା

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