 Mathematics 9 (Complete Solutions)

# Q4 Verify that

If
A =

 1 2 4 6

,
B =

 3 -1 2 -2

, then verify that.

 (i) A(Adj A) = (Adj A) A = (∣ A ∣)I (ii) BB-1 = I = B-1B

Solution:

First we will find A(Adj A)

 6 -2 -4 1

So,

 1 2 4 6

 6 -2 -4 1

 1(6) + 2(-4) 1(-2) + 2(1) 4(6) + 6(-4) 4(-2) + 6(1)

 6 - 8 -2 + 2 24 - 24 -8 + 6

 -2 0 0 -2

---------- ( i )
Now we will find (Adj A)A

 6 -2 -4 1

 1 2 4 6

 6(1) + (-2)(4) 6(2) + (-2)(6) -4(1) + 1(4) -4(2) + 1(6)

 6 - 8 12 - 12 -4 + 4 -8 + 6

 -2 0 0 -2

---------- ( ii )
Now we will find (∣ A ∣)I
∣ A ∣ =

 1 2 4 6

= 1(6) - 2(4)
∣ A ∣ = 6 - 8
∣ A ∣ = -2
So,
(∣ A ∣)I = -2

 1 0 0 1

(∣ A ∣)I =

 -2(1) -2(0) (-2)(0) (-2)(1)

(∣ A ∣)I =

 -2 0 0 -2

---------- ( iii )
Hence from equations ( i ), ( ii ) and ( iii ) it is verified that A(Adj A) = (Adj A) A = (∣ A ∣)I

(ii) BB-1 = I = B-1B
First we will find BB-1
B-1 =
Now we will find ∣ B ∣
∣ B ∣ =

 3 -1 2 -2

= (3)(-2) - (-1)(2)
∣ B ∣ = -6 + 2
∣ B ∣ = -4
Now we will find Adj B

 -2 1 -2 3

So,
B-1 =

 -2 1 -2 3

-4
B-1 =
-
 1 4

 -2 1 -2 3

B-1 =

-2 (-
 1 4
)
1 (-
 1 4
)
-2 (-
 1 4
)
3 (-
 1 4
)

B-1 =

 2 4
-
 1 4
 2 4
-
 3 4

B-1 =

 1 2
-
 1 4
 1 2
-
 3 4

So,
BB-1 =

 3 -1 2 -2

 1 2
-
 1 4
 1 2
-
 3 4

BB-1 =

3
(
 1 2
)
+ (-1)
(
 1 2
)
3
( -
 1 4
)
+ (-1)
( -
 3 4
)
2
(
 1 2
)
+ (-2)
(
 1 2
)
2
( -
 1 4
)
+ (-2)
( -
 3 4
)

BB-1 =

 3 2
-
 1 2
-
 3 4
+
 3 4
 2 2
-
 2 2
-
 2 4
+
 6 4

BB-1 =

 3 2
-
 1 2
-
 3 4
+
 3 4
1 - 1
-
 1 2
+
 3 2

BB-1 =

 3 - 1 2
 -3 + 3 4
0
 3 - 1 2

BB-1 =

 2 2
 0 4
0
 2 2

BB-1 =

 1 0 0 1

You know that I =

 1 0 0 1

Now we will find B-1B
B-1B =

 1 2
-
 1 4
 1 2
-
 3 4

 3 -1 2 -2

BB-1 =

 1 2
(3) +
( -
 1 4
)
(2)
 1 2
(-1) +
( -
 1 4
)
(-2)
 1 2
(3) +
( -
 3 4
)
(2)
 1 2
(-1) +
( -
 3 4
)
(-2)

BB-1 =

 3 2
-
 2 4
-
 1 2
+
 2 4
 3 2
-
 6 4
-
 1 2
+
 6 4

BB-1 =

 3 2
-
 1 2
-
 1 2
+
 1 2
 3 2
-
 3 2
-
 1 2
+
 3 2

BB-1 =

 3 - 1 2
 -1 + 1 2
 3 - 3 2
 -1 + 3 2

BB-1 =

 2 2
 0 2
 0 2
 2 2

BB-1 =

 1 0 0 1

Hence it is verified that BB-1 = I = B-1B