Let R=\(\begin{pmatrix}a&3&b\\c&2&d\\0&5&0\end{pmatrix}\): a,b,c,d ∈ {0,3,5,7,11,13,17,19}. Then the number of invertible matrices in R is
Approach Solution - 1
The absolute value of matrix R is negative 5, and the absolute value of matrix P is also negative 5. The determinant ∣R∣ can be equal to zero in the following cases:
(i) Two of the values a, b, c, d are zeros, which can be (a and b), (b and d), (d and c), or (c and a) \(→\) 4 × 72 ways = 196.
(ii) Any three of the values a, b, c, d are zeros \(→\) 4C3×7=28.
(iii) All four of the values a, b, c, d are zeros \(→\) 1.
(iv) All four of the values a, b, c, d are non-zero but the same number \(→\) 7.
(v) When two are alike and the other two are alike (non-zero) \(→\) 7C2×2×2=84.
The number of invertible matrices = 84 – 196 – 28 – 1 – 7 – 84 = 3780.
So, the answer is 3780.
Approach Solution -2
Given :
Total matrices = 84 = 4096
|R| = 5(bc – ad)
No. of non-invertible matrices : bc = ad
Case I :
If a, b, c, d ≠ 0, then cases = 7C2 . 2! 2! + 7C1(1) = 91
Case II :
if ad = bc = 0, then cases = 15C1 . 15C1 = 225
( ad & bc can take any combination from 0 × 0, 0 × 3, 0 × 5, ……, 0 × 19, 3 × 0, 5 × 0, ….. 19 × 0)
Number of invertible matrices :
= 4096 – (91 + 225)
= 3780
So, the correct answer is 3780.
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The graph of a linear inequality in one variable is a number line. We can use an open circle for < and > and a closed circle for ≤ and ≥.
Inequalities that have the same solution are commonly known as equivalents. There are several properties of inequalities as well as the properties of equality. All the properties below are also true for inequalities including ≥ and ≤.
The addition property of inequality says that adding the same number to each side of the inequality gives an equivalent inequality.
If x>y, then x+z>y+z If x>y, then x+z>y+z
If x<y, then x+z<y+z If x<y, then x+z<y+z
The subtraction property of inequality tells us that subtracting the same number from both sides of an inequality produces an equivalent inequality.
If x>y, then x−z>y−z If x>y, then x−z>y−z
If x<y, then x−z<y−z Ifx<y, then x−z<y−z
The multiplication property of inequality tells us that multiplication on both sides of an inequality with a positive number gives an equivalent inequality.
If x>y and z>0, then xz>yz If x>y and z>0, then xz>yz
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