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Matrices

Long Answer Type

191.

If $A = |\begin{matrix} 2 & - 3 & 5 \\ 3 & 2 & - 4 \\ 1 & 1 & - 2 \end{matrix}|, f i n d A^{- 1} .$ Use it solve the system of equations.

2x - 3y + 5z = 11

3x + 2y - 4z = -5

x + y- 2z =-3

192.

Using elementary row transformations, find the inverse of the matrix $A = [\begin{matrix} 1 & 2 & 3 \\ 2 & 5 & 7 \\ - 2 & - 4 & - 5 \end{matrix}]$

Short Answer Type

193.

Find the value of x and y if: $2 [\begin{matrix} 1 & 3 \\ 0 & x \end{matrix}] + [\begin{matrix} y & 0 \\ 1 & 2 \end{matrix}] = [\begin{matrix} 5 & 6 \\ 1 & 8 \end{matrix}]$

Long Answer Type

194.
Let $A = |\begin{matrix} 3 & 2 & 5 \\ 4 & 1 & 3 \\ 0 & 6 & 7 \end{matrix}| .$ Express A as sum of two matrices such that one is symmetric and the other is skew symmetric.

$A = |\begin{matrix} 3 & 2 & 5 \\ 4 & 1 & 3 \\ 0 & 6 & 7 \end{matrix}| A' = |\begin{matrix} 3 & 4 & 0 \\ 2 & 1 & 6 \\ 5 & 3 & 7 \end{matrix}|$

Now, A can be written as:

$A = \frac{1}{2} A + A' + \frac{1}{2} A = A' A + A' = |\begin{matrix} 3 & 2 & 5 \\ 4 & 1 & 3 \\ 0 & 6 & 7 \end{matrix}| + |\begin{matrix} 3 & 4 & 0 \\ 2 & 1 & 6 \\ 5 & 3 & 7 \end{matrix}| = |\begin{matrix} 3 + 3 & 2 + 4 & 5 + 0 \\ 4 + 2 & 1 + 1 & 3 + 6 \\ 0 + 5 & 6 + 3 & 7 + 7 \end{matrix}| = |\begin{matrix} 6 & 6 & 5 \\ 6 & 2 & 9 \\ 5 & 9 & 14 \end{matrix}| \frac{1}{2} A + A' = \frac{1}{2} |\begin{matrix} 6 & 6 & 5 \\ 6 & 2 & 9 \\ 5 & 9 & 14 \end{matrix}| = |\begin{matrix} 3 & 3 & \frac{5}{2} \\ 3 & 1 & \frac{9}{2} \\ \frac{5}{2} & \frac{9}{2} & 7 \end{matrix}| = P, say Now, P' = |\begin{matrix} 3 & 3 & \frac{5}{2} \\ 3 & 1 & \frac{9}{2} \\ \frac{5}{2} & \frac{9}{2} & 7 \end{matrix}| Thus, P = \frac{1}{2} A + A' is a symmetric matrix .$

$Now, A - A' = |\begin{matrix} 3 - 3 & 2 - 4 & 5 - 0 \\ 4 - 2 & 1 - 1 & 3 - 6 \\ 0 - 5 & 6 - 3 & 7 - 7 \end{matrix}| = |\begin{matrix} 0 & - 2 & 5 \\ 2 & 0 & - 3 \\ - 5 & 3 & 0 \end{matrix}| \frac{1}{2} A - A' = |\begin{matrix} 0 & - 1 & \frac{5}{2} \\ 1 & 0 & \frac{- 3}{2} \\ \frac{- 5}{2} & \frac{3}{2} & 0 \end{matrix}| = Q, say Now, Q' = |\begin{matrix} 0 & 1 & \frac{- 5}{2} \\ - 1 & 0 & \frac{3}{2} \\ \frac{5}{2} & \frac{- 3}{2} & 0 \end{matrix}| = - |\begin{matrix} 0 & - 1 & \frac{- 5}{2} \\ 1 & 0 & \frac{- 3}{2} \\ \frac{- 5}{2} & \frac{3}{2} & 0 \end{matrix}| = - Q Thus, Q = \frac{1}{2} A - A' is a skew symmetric matrix . ∴ A = |\begin{matrix} 3 & 2 & 5 \\ 4 & 1 & 3 \\ 0 & 6 & 7 \end{matrix}| = |\begin{matrix} 3 & 3 & \frac{5}{2} \\ 3 & 1 & \frac{9}{2} \\ \frac{5}{2} & \frac{- 3}{2} & 7 \end{matrix}| + |\begin{matrix} 0 & - 1 & \frac{- 5}{2} \\ 1 & 0 & \frac{- 3}{2} \\ \frac{- 5}{2} & \frac{3}{2} & 0 \end{matrix}|$

195.

If A = $[\begin{matrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{matrix}]$ , verify that A² – 4A – 5I = 0.

Short Answer Type

196.

If A is an invertible matrix of order 3 and |A| = 5, then find |adj. A|.

197.

If matrix A = (1, 2, 3), write AA’, where A’ is the transpose of matrix A.

Long Answer Type

198.

Using matrices, solve the following system of equations:
2x – 3y + 5 = 11
3x + 2y – 4z = -5
x + y – 2z= -3

Short Answer Type

199.

If A = $[\begin{matrix} cosα & - sinα \\ sinα & cosα \end{matrix}]$ , then for what value of α is A an identity matrix?

200.

If $[\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] [\begin{matrix} 3 & 1 \\ 2 & 5 \end{matrix}] = [\begin{matrix} 7 & 11 \\ k & 23 \end{matrix}]$ , then write the value of k.