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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.

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

2x - 3y + 5z = 11

3x + 2y - 4z = -5

x + y - 2z = -3

System of equations can be written as AX = B

$Where, A = [\begin{matrix} 2 & - 3 & 5 \\ 3 & 2 & - 4 \\ 1 & 1 & - 2 \end{matrix}] X = [\begin{matrix} x \\ y \\ z \end{matrix}] B = [\begin{matrix} 11 \\ - 5 \\ - 3 \end{matrix}] A = [\begin{matrix} 2 & - 3 & 5 \\ 3 & 2 & - 4 \\ 1 & 1 & - 2 \end{matrix}] |A| = 2 (- 4 + 4) + 3 (- 6 + 4) + 5 (3 - 2) |A| = - 6 + 5 = - 1 \neq 0 ∴ A^{- 1} exists and system of equation has unique solution A^{- 1} = \frac{1}{|A|} (adj A)$

$adj A = [\begin{matrix} 0 & - 1 & 2 \\ 2 & - 9 & 23 \\ 1 & - 5 & 13 \end{matrix}] A^{- 1} = \frac{1}{|A|} [\begin{matrix} 0 & - 1 & 2 \\ 2 & - 9 & 23 \\ 1 & - 5 & 13 \end{matrix}] = [\begin{matrix} 0 & 1 & - 2 \\ - 2 & 9 & - 23 \\ - 1 & 5 & - 13 \end{matrix}] X = A^{- 1} B = [\begin{matrix} 0 & 1 & - 2 \\ - 2 & 9 & - 23 \\ - 1 & 5 & - 13 \end{matrix}] [\begin{matrix} 11 \\ - 5 \\ - 3 \end{matrix}] X = [\begin{matrix} - 5 + 6 \\ - 22 - 45 + 69 \\ - 11 - 25 + 39 \end{matrix}] = [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$

So, x = 1, y = 2, z = 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.