Electronics and Communication Engineering (ECE) Exam > Electronics and Communication Engineering (ECE) Questions > Let A be a 3 x 3 matrix with eigen values 1,...
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Let A be a 3 x 3 matrix with eigen values 1, -1 and 3. Then
- a)A2 + A is non - singular
- b)A2 - A is non - singular
- c)A2 + 3A is non - singular
- d)A2 - 3A is non - singular
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
Let A be a 3 x 3 matrix with eigen values 1, -1 and 3. Thena)A2 + A ...
Explanation:
To solve this problem, we need to understand the properties of eigenvalues and eigenvectors of a matrix.
Eigenvalues:
Eigenvalues are the values λ for which the equation Ax = λx has non-zero solutions. In this case, the eigenvalues of matrix A are given as 1, -1, and 3.
Eigenvectors:
Eigenvectors are the non-zero vectors x that satisfy the equation Ax = λx. The eigenvectors corresponding to eigenvalues 1, -1, and 3 are denoted as x1, x2, and x3 respectively.
Now let's consider the given options:
a) A^2 is non-singular:
A^2 represents the matrix obtained by multiplying matrix A with itself. If A^2 is non-singular, it means that the determinant of A^2 is non-zero.
However, the determinant of A^2 is equal to the product of the eigenvalues of A raised to the power of 2. In this case, the eigenvalues of A are 1, -1, and 3. So, the determinant of A^2 is (1^2) * (-1^2) * (3^2) = 9. Since the determinant is non-zero, A^2 is non-singular. This option is correct.
b) A^2 - A is non-singular:
Similarly, we can calculate the determinant of A^2 - A. However, this determinant will be zero because the eigenvalues are subtracted from the diagonal elements in A^2 - A. Therefore, A^2 - A is singular. This option is incorrect.
c) A^2 + 3A is non-singular:
Here, we need to calculate the determinant of A^2 + 3A. The determinant is equal to the product of the eigenvalues of A raised to the power of 2, multiplied by the eigenvalues of A raised to the power of 3. In this case, the determinant is (1^2) * (-1^2) * (3^2) * (1^3) * (-1^3) * (3^3) = 0. Since the determinant is zero, A^2 + 3A is singular. This option is incorrect.
d) A^2 - 3A is non-singular:
Similarly, we can calculate the determinant of A^2 - 3A. In this case, the determinant is (1^2) * (-1^2) * (3^2) * (1^3) * (-1^3) * (3^3) = 0. Since the determinant is zero, A^2 - 3A is singular. This option is incorrect.
Therefore, the correct answer is option 'C': A^2 + 3A is non-singular.
To solve this problem, we need to understand the properties of eigenvalues and eigenvectors of a matrix.
Eigenvalues:
Eigenvalues are the values λ for which the equation Ax = λx has non-zero solutions. In this case, the eigenvalues of matrix A are given as 1, -1, and 3.
Eigenvectors:
Eigenvectors are the non-zero vectors x that satisfy the equation Ax = λx. The eigenvectors corresponding to eigenvalues 1, -1, and 3 are denoted as x1, x2, and x3 respectively.
Now let's consider the given options:
a) A^2 is non-singular:
A^2 represents the matrix obtained by multiplying matrix A with itself. If A^2 is non-singular, it means that the determinant of A^2 is non-zero.
However, the determinant of A^2 is equal to the product of the eigenvalues of A raised to the power of 2. In this case, the eigenvalues of A are 1, -1, and 3. So, the determinant of A^2 is (1^2) * (-1^2) * (3^2) = 9. Since the determinant is non-zero, A^2 is non-singular. This option is correct.
b) A^2 - A is non-singular:
Similarly, we can calculate the determinant of A^2 - A. However, this determinant will be zero because the eigenvalues are subtracted from the diagonal elements in A^2 - A. Therefore, A^2 - A is singular. This option is incorrect.
c) A^2 + 3A is non-singular:
Here, we need to calculate the determinant of A^2 + 3A. The determinant is equal to the product of the eigenvalues of A raised to the power of 2, multiplied by the eigenvalues of A raised to the power of 3. In this case, the determinant is (1^2) * (-1^2) * (3^2) * (1^3) * (-1^3) * (3^3) = 0. Since the determinant is zero, A^2 + 3A is singular. This option is incorrect.
d) A^2 - 3A is non-singular:
Similarly, we can calculate the determinant of A^2 - 3A. In this case, the determinant is (1^2) * (-1^2) * (3^2) * (1^3) * (-1^3) * (3^3) = 0. Since the determinant is zero, A^2 - 3A is singular. This option is incorrect.
Therefore, the correct answer is option 'C': A^2 + 3A is non-singular.
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Community Answer
Let A be a 3 x 3 matrix with eigen values 1, -1 and 3. Thena)A2 + A ...
Singular Matrices have Zero Eigenvalues. Suppose A is a square matrix. Then A is singular if and only if λ=0 is an eigenvalue of A.
Eigen values ofA2+ A are 2,0,12
Eigen values of A2 - A are 0,2,6,
Eigen values of A2 +3A are 4,-2,18
Eigen values of A2 - 3A are -2,4,0.
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Let A be a 3 x 3 matrix with eigen values 1, -1 and 3. Thena)A2 + A is non - singularb)A2 - A is non - singularc)A2 + 3A is non - singulard)A2 - 3A is non - singularCorrect answer is option 'C'. Can you explain this answer?
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