Matrices and Determinants: Matrix Operations, Rank, Inverse, Systems of Linear Equations and Eigenvalues

Let A and B be two matrices and gamma be any scalar, then :

Let A and B be two invertible matrices, then

which of the following is true ?

Let A = [[1, 2, 0, -1], [2, 6, -3, -3], [3, 10, -6, -5]], then rank of matrix A is :

Let W1 and W2 are two subspaces of a vector space V over the field F, then :

Let B = [[5, -9, 6], [0, 2, 3], [0, 0, 7]], then its reduced echelon form is :

Let $\mathbb{R}^n = V$ be a vector space over the field of real numbers, then

which of the following is a subspace ?

A toy manufacturer produces an inexpensive doll (flopsy) and an expensive doll (mopsy) in units of $x$ hundreds and $y$ hundreds respectively. Suppose it is possible to produce the doll in such a way that $y = \frac{82-10x}{10-x}, 0 \le x \le 8$ and that the company receives twice as much for selling a mopsy doll as for selling a flopsy doll. The level of production for both flopsy and mopsy dolls for which the total revenue derived from selling these dolls is maximized is given by :

The determinant of the $7 \times 7$ anti-diagonal matrix $A$ having $1$s on the anti-diagonal and $0$s elsewhere is:

Let $T: \mathbb{R}^2 \to \mathbb{R}^2$ be a linear transformation such that $T$ is a rotation by an angle of $90^\circ$ in the clockwise direction in $\mathbb{R}^2$. Then:

Let $A$ be an $n \times n$ matrix of real numbers such that $A^2 = I$. Then the eigenvalues of $A$ are:

Let $A$ be a $5 \times 5$ skew-symmetric matrix of real numbers. Then

which of the following is true?

Which of the following statements is always true?

Let $V$ and $W$ be vector spaces and $T: V \to W$ be a linear transformation. Then:

The inverse of matrix $A = \begin{pmatrix} 0 & 2 & 4 \\ 2 & 4 & 2 \\ 3 & 3 & 1 \end{pmatrix}$ is:

Let $A_{m \times n}$ be any matrix and $\operatorname{Rank}(A) = 0$. Then matrix $A$ is:

Which of the following statements is always true for a square matrix?

If $A$ and $B$ be orthogonal matrices of same size, then :