SATHEE: Exercise 04

04 ਨਿਰਧਾਰਕ

ਅਭਿਆਸ 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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03 ਮੈਟ੍ਰਿਕਸ

04 ਨਿਰਧਾਰਕ

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