Practicing the Intermediate 1st Year Maths 1A Textbook Solutions Inter 1st Year Maths 1A Matrices Solutions Exercise 3(f) will help students to clear their doubts quickly.

## Intermediate 1st Year Maths 1A Matrices Solutions Exercise 3(f)

I. Find the rank of each of the following matrices.

Question 1.

\(\left[\begin{array}{ll}

1 & 0 \\

0 & 0

\end{array}\right]\)

Solution:

Det A = \(\left|\begin{array}{ll}

1 & 0 \\

0 & 0

\end{array}\right|\) = 0 – 0 = 0

and |1| = 1 ≠ 0

∴ ρ(A) = 1

Question 2.

\(\left[\begin{array}{ll}

1 & 0 \\

0 & 1

\end{array}\right]\)

Solution:

Det A = \(\left|\begin{array}{ll}

1 & 0 \\

0 & 1

\end{array}\right|\) = 1 – 0 = 1 ≠ 0

∴ ρ(A) = 2

Question 3.

\(\left[\begin{array}{ll}

1 & 1 \\

0 & 0

\end{array}\right]\)

Solution:

Det A = \(\left|\begin{array}{ll}

1 & 1 \\

0 & 0

\end{array}\right|\) = 0 – 0 = 0

|1| = 1 ≠ 0

∴ ρ(A) = 1

Question 4.

\(\left[\begin{array}{ll}

1 & 1 \\

1 & 0

\end{array}\right]\)

Solution:

Det A = \(\left|\begin{array}{ll}

1 & 1 \\

1 & 0

\end{array}\right|\) = 0 – 1 = -1 ≠ 0

∴ ρ(A) = 2

Question 5.

\(\left[\begin{array}{ccc}

1 & 0 & -4 \\

2 & -1 & 3

\end{array}\right]\)

Solution:

\(\left|\begin{array}{cc}

1 & -4 \\

2 & 3

\end{array}\right|\) = 3 + 8 = -11 ≠ 0

∴ ρ(A) = 2

Question 6.

\(\left[\begin{array}{lll}

1 & 2 & 6 \\

2 & 4 & 3

\end{array}\right]\)

Solution:

\(\left|\begin{array}{ll}

2 & 6 \\

4 & 3

\end{array}\right|\) = 6 – 24 = -18 ≠ 0

∴ ρ(A) = 2

II.

Question 1.

\(\left[\begin{array}{lll}

1 & 0 & 0 \\

0 & 0 & 1 \\

0 & 1 & 0

\end{array}\right]\)

Solution:

Det A = \(\left|\begin{array}{lll}

1 & 0 & 0 \\

0 & 1 & 0 \\

0 & 0 & 1

\end{array}\right|\)

= 1(1 – 0) – 0(0 – 0) + 0(0 – 0)

= 1 – 0 + 0

= 1 ≠ 0

∴ ρ(A) = 3

Question 2.

\(\left[\begin{array}{ccc}

1 & 4 & -1 \\

2 & 3 & 0 \\

0 & 1 & 2

\end{array}\right]\)

Solution:

Det A = \(\left|\begin{array}{ccc}

1 & 4 & -1 \\

2 & 3 & 0 \\

0 & 1 & 2

\end{array}\right|\)

= 1(6 – 0) – 2(8 + 1) + 0(0 + 3)

= 6 – 18

= -12 ≠ 0

∴ ρ(A) = 3

Question 3.

\(\left[\begin{array}{lll}

1 & 2 & 3 \\

2 & 3 & 4 \\

0 & 1 & 2

\end{array}\right]\)

Solution:

Det A = \(\left|\begin{array}{lll}

1 & 2 & 3 \\

2 & 3 & 4 \\

0 & 1 & 2

\end{array}\right|\)

= 1(6 – 4) – 2(4 – 3) + 0(8 – 9)

= 2 – 2 + 0

= 0

∴ ρ(A) ≠ 3, ρ(A) < 3

Take \(\left|\begin{array}{ll}

1 & 2 \\

2 & 3

\end{array}\right|\) = 3 – 4 = -1 ≠ 0

∴ ρ(A) = 2

Question 4.

\(\left[\begin{array}{lll}

1 & 1 & 1 \\

1 & 1 & 1 \\

1 & 1 & 1

\end{array}\right]\)

Solution:

Let A = \(\left[\begin{array}{lll}

1 & 1 & 1 \\

1 & 1 & 1 \\

1 & 1 & 1

\end{array}\right]\), det A = 0, ρ(A) ≠ 3

All 2 × 2 sub-matrix det. is zero

∴ ρ(A) ≠ 2

|1| = 1 ≠ 0

∴ ρ(A) = 1

Question 5.

\(\left[\begin{array}{cccc}

1 & 2 & 0 & -1 \\

3 & 4 & 1 & 2 \\

-2 & 3 & 2 & 5

\end{array}\right]\)

Solution:

Take sub-matrix B = \(\left|\begin{array}{ccc}

1 & 2 & 0 \\

3 & 4 & 1 \\

-2 & 3 & 2

\end{array}\right|\)

= 1(8 – 3) – 2(6 + 2)

= 5 – 16

= -11 ≠ 0

Rank of the given matrix is 3.

Question 6.

\(\left[\begin{array}{cccc}

0 & 1 & 1 & -2 \\

4 & 0 & 2 & 5 \\

2 & 1 & 3 & 1

\end{array}\right]\)

Solution:

Take sub matrix A = \(\left[\begin{array}{lll}

0 & 1 & 1 \\

4 & 0 & 2 \\

2 & 1 & 3

\end{array}\right]\)

= -1(12 – 4) + 1(4 – 0)

= -8 + 4

= -4 ≠ 0

∴ ρ(A) = 3