[amp_mcq option1=”0″ option2=”1″ option3=”2″ option4=”3″ correct=”option3″]<\/p>\n
The minimum eigenvalue of the matrix $A$ is $\\lambda_1 = -2.5$.<\/p>\n
To find the eigenvalues of a matrix, we can use the following formula:<\/p>\n
$$\\lambda_i = \\frac{1}{|A – \\lambda_i I|}$$<\/p>\n
where $I$ is the identity matrix and $|A|$ is the determinant of $A$.<\/p>\n
In this case, we have:<\/p>\n
$$|A – \\lambda_i I| = \\begin{vmatrix} 3 – \\lambda_i & 5 & 2 \\ 5 & 12 – \\lambda_i & 7 \\ 2 & 7 & 5 – \\lambda_i \\end{vmatrix}$$<\/p>\n
Expanding the determinant, we get:<\/p>\n
$$|A – \\lambda_i I| = -\\lambda_i^3 + 30 \\lambda_i^2 – 257 \\lambda_i + 1260$$<\/p>\n
We can then use the quadratic formula to solve for the eigenvalues:<\/p>\n
$$\\lambda_i = \\frac{-30 \\pm \\sqrt{30^2 – 4 \\cdot (-1) \\cdot 1260}}{2 \\cdot (-1)}$$<\/p>\n
$$\\lambda_i = \\frac{-30 \\pm \\sqrt{12960}}{-2}$$<\/p>\n
$$\\lambda_i = \\frac{-30 \\pm 113}{-2}$$<\/p>\n
$$\\lambda_i = -2.5 \\pm 56.5$$<\/p>\n
Therefore, the minimum eigenvalue of $A$ is $\\lambda_1 = -2.5$.<\/p>\n","protected":false},"excerpt":{"rendered":"
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