Referring to the AP Inter 2nd Year Maths Study Material Chapter 4 Determinants Exercise 4c Solutions makes it easier to understand complex problems.
AP Inter 2nd Year Maths Determinants Solutions Exercise 4c
I.
Question 1.
Write Minors and Cofactors of the elements of \(\left|\begin{array}{cc}
2 & -4 \\
0 & 3
\end{array}\right|\)
Solution:
The given determinant is \(\left|\begin{array}{cc}
2 & -4 \\
0 & 3
\end{array}\right|\)
Minor means little determinant
Minor of the element ajj is Mjj
Here a11 = 1. So M11 = Minor of a11 = 3
M11 = Minor of the element a11 = 3; M12 = Minor of the element a12 = 0;
M21 = Minor of the element a21 = -4; M22 = Minor of the element a22 = 2;
Now, cofactor of aij is Aij = (-1)i+j Mij
A11 =(-1)1 + 1(3) = 3;
A12 =(-1)1 + 2 (0) = 0;
A21 = (-1)2 + 1 (-4) = 4;
A22 = (-1)2 + 2 (2) = 2
Question 2.
Write Minors and Cofactors of the elements of \(\left|\begin{array}{ll}
a & c \\
b & d
\end{array}\right|\)
Solution:
The given determinant is \(\left|\begin{array}{ll}
a & c \\
b & d
\end{array}\right|\)
Minor of the element ajj is Mjj
M11 = Minor of the element a11 = d; M12 = Minor of the element a12 = b;
M21 = Minor of the element a21 = c; M22 = Minor of the element a22 = a;
Now, cofactor of ajj is Ajj = (-1)i + j Mjj
A11 = (-1)1 +1 (d) = d; A12 = (-1)1+2 (b) = -b
A21 = (-1 )2 + 1 (c) = -c; A22 = (-1)2 + 2 (a) = a
Question 3.
Using Cofactors of elements of second row, evaluate ∆ = \(\left|\begin{array}{lll}
5 & 3 & 8 \\
2 & 0 & 1 \\
1 & 2 & 3
\end{array}\right|\)
Solution:
The given determinant is \(\left|\begin{array}{lll}
5 & 3 & 8 \\
2 & 0 & 1 \\
1 & 2 & 3
\end{array}\right|\)
Minor of the element ajj is Mjj
M21 = Minor of the element a21 = \(\left|\begin{array}{ll}
3 & 8 \\
2 & 3
\end{array}\right|\) = (3 × 3) – (8 × 2) = -7
A21 = (-1)2+1 (-7) = 7
M22 = Minor of the element a22 = \(\left|\begin{array}{ll}
5 & 8 \\
1 & 3
\end{array}\right|\) = (5 × 3) – (8 × 1) = 15 – 8 = 7
A22 = (-1)2+2 (7) = 7
M23 Minor of the element a23 = \(\left|\begin{array}{ll}
5 & 3 \\
1 & 2
\end{array}\right|\) = (5 × 2) – (3 × 1) = 10 – 3 = 7
A23 = (-1)2 + 3 (7) = -7
We know that ∆ is equal to the sum of the product of the elements of the second row with their corresponding cofactors.
∆ = a21A21 + a22A22 + a23A23
= 2(7) + 0(7) + 1(-7) = 14 – 7 = 7
Question 4.
Using Cofactors of elements of third column, evaluate ∆ = \(\left|\begin{array}{ccc}
1 & x & y z \\
1 & y & z x \\
1 & z & x y
\end{array}\right|\)
Solution:
The given determinant is \(\left|\begin{array}{ccc}
1 & x & y z \\
1 & y & z x \\
1 & z & x y
\end{array}\right|\)
M13 = \(\left|\begin{array}{ll}
1 & y \\
1 & z
\end{array}\right|\) (1 × z) – (1 × y) = z – y; A13 = (-1)4(z – y) = zy
M23 =\(\left|\begin{array}{ll}
1 & x \\
1 & z
\end{array}\right|\) = 1 × Z – x × 1 = z – x; A23 = (-1)5(z – x) = -(z – x) = x – z
M33 = \(\left|\begin{array}{ll}
1 & x \\
1 & y
\end{array}\right|\) =1 × y – x × 1 = y – x. A33 = (-1)6(y – x) = y – x
We know that ∆ is equal to the sum of the product of the elements of the second row
with their corresponding cofactors.
∆ = a13A13 + a23A23 + a33A33 .
= yz(z – y) + zx(x – z) + xy(y – x) = yz2 – y2z + x2z – xz2 + xy2 – x2y
=(x2z – y2z) + (yz2 – xz2) + (xy2 – x2y) = z(x2 – y2) + z2(y – x) + xy(y – x)
= z(x – y)(x + y) + z2 (y – x) + xy(y – x) = (x – y)[zx – z2 + zy – xy]
= (x – y)[z(x – z) + y(z – x)} =(x – y)(z – x)[-z + y]
= (x – y)(y – z)(z – x)
∴ ∆ = (x – y)(y – z)(z – x)
II.
Question 1.
Find minors and cofactors of the elements of the determinant \(\left|\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right|\)
Solution:
The given determinant is \(\left|\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right|\). Minor of the element aij is Mij
Minor of the elements a11 is M11 = \(\left|\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right|\) = 1; M12 = \(\left|\begin{array}{ll}
0 & 0 \\
0 & 1
\end{array}\right|\) = 0; M13 = \(\left|\begin{array}{ll}
0 & 1 \\
0 & 0
\end{array}\right|\) = 0
Similarly, M21 = \(\left|\begin{array}{ll}
0 & 0 \\
0 & 1
\end{array}\right|\) = 0; M22 = \(\left|\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right|\) = 1; M23 = \(\left|\begin{array}{ll}
1 & 0 \\
0 & 0
\end{array}\right|\) = 0
M31 = \(\left|\begin{array}{ll}
0 & 0 \\
0 & 1
\end{array}\right|\) = 0; M31 = \(\left|\begin{array}{ll}
1 & 0 \\
0 & 0
\end{array}\right|\) = 0; M33 = \(\left|\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right|\) = 1
Now, cofactor of aij is Aij = (-1)i + j Mij
A11 = (-1)1 + 1 (1) = 1; A12 =(-1)1 + 2(0) = 0; A13 =(-1)1 + 3(0) = 0;
A21 = (-1)2 + 1 (0) = 0; A22 = (-1)2 + 2(1) = 1; A23 = (-1)2 + 3(0) = 0
A31 = (-1)3 + 1 (0) = 0; A32 = (-1)3 + 2(0) = 0; A33 = (-1)3 + 3 (1) = 1
Question 2.
Find minors and cofactors of the elements of the determinant \(\left|\begin{array}{ccc}
1 & 0 & 4 \\
3 & 5 & -1 \\
0 & 1 & 2
\end{array}\right|\)
Solution:
The given determinant is \(\left|\begin{array}{ccc}
1 & 0 & 4 \\
3 & 5 & -1 \\
0 & 1 & 2
\end{array}\right|\). Minor of the element aij is Mij
Minor of the element a11 is M11 = \(\left|\begin{array}{cc}
5 & -1 \\
1 & 2
\end{array}\right|\) = (5 × 2) – (1 × -1) = 10 + 1 = 11
M12 = \(\left|\begin{array}{cc}
3 & -1 \\
0 & 2
\end{array}\right|\) = (3 × 2) – (1 × 0) = 6;
M13 = \(\left|\begin{array}{cc}
3 & 5 \\
0 & 1
\end{array}\right|\) = (3 × 1) – (5 × 0) = 3
M21 = \(\left|\begin{array}{cc}
0 & 4 \\
1 & 2
\end{array}\right|\) (0 × 2) – (4 × 1) = 4; M22 = \(\left|\begin{array}{cc}
1 & 4 \\
0 & 2
\end{array}\right|\) = (1 × 2) – (4 × 0) = 2
M23 = \(\left|\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}\right|\) = (1 × 1) – (0 × 0) = 1;
M31 = \(\left|\begin{array}{cc}
0 & 4 \\
5 & -1
\end{array}\right|\) = (0 × -1)- (4 × 5) = -20. M32 = \(\left|\begin{array}{cc}
1 & 4 \\
3 & -1
\end{array}\right|\) =(1 × -1) – (4 × 3) = -13
M33 = \(\left|\begin{array}{cc}
1 & 0 \\
3 & 5
\end{array}\right|\) = (1 × 5) – (0 × 3) = 5
Now, cofactor of aij is Aij = (-1)sup>i + j Mij
A11 =(-1)1 + 1(11) = 11; A12 = (-1)1 + 2 (6) = -6; A13 = (-1)1 + 3 (3) = 3
A21 = (-1)2 + 1 (-4) = 4; A22 = (-1)2 + 2 (2) = 2; A23 = (-1)2 + 3 (1) = -1
A31 =(-1)3 + 1 (-20) = -20; A32 = (-1)3 + 2 (13)=13; A33 = (-1)3 + 3 (5) = 5