Let A be an n ×n matrix. (a) Show that if the entries of some row of 𝐀 are proportional to those

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Let A be an n ×n matrix. (a) Show that if the entries of...

Let $\mathrm{A}$ be an $n \times n$ matrix. (a) Show that if the entries of some row of $\mathbf{A}$ are proportional to those in another row, then $|\mathbf{A}|=0$. (b) Show that if the entries in every row of $\mathbf{A}$ add up to zero, then $|\mathbf{A}|=0$. (Hint: Consider the system $\mathbf{A X}=\mathbf{0}$, and note that the $n \times 1$ vector $\mathbf{X}$ having every entry equal to 1 is a nontrivial solution.)

Let $\mathrm{A}$ be an $n \times n$ matrix.
(a) Show that if the entries of some row of $\mathbf{A}$ are proportional to those in another row, then $|\mathbf{A}|=0$.
(b) Show that if the entries in every row of $\mathbf{A}$ add up to zero, then $|\mathbf{A}|=0$. (Hint: Consider the system $\mathbf{A X}=\mathbf{0}$, and note that the $n \times 1$ vector $\mathbf{X}$ having every entry equal to 1 is a nontrivial solution.)

Key Concepts

Linear Dependence

Linear dependence refers to a situation where one row (or column) of a matrix can be expressed as a scalar multiple or a combination of other rows. In the context of determinants, if a row is proportional to another, they do not provide independent information, causing the determinant to vanish. This property is fundamental in understanding why certain singular matrices have a zero determinant.

Properties of Determinants

Determinants encode various geometric and algebraic properties of matrices. One key property is that if any two rows (or columns) of a matrix are linearly dependent, the determinant is zero. This is because the determinant represents a volume scaling factor, and linear dependence collapses the volume to zero. Additionally, determinants are used to assess the invertibility of a matrix: a non-zero determinant indicates an invertible matrix, while a zero determinant indicates a singular matrix.

Kernel (Null Space) of a Matrix

The kernel or null space of a matrix consists of all vectors that, when multiplied by the matrix, yield the zero vector. If there exists a nontrivial solution (i.e., a solution other than the zero vector) to Ax = 0, it implies that the matrix is singular and its determinant must be zero. For example, if every row of a matrix sums to zero, the vector with all entries equal to one is in the kernel, revealing the presence of linear dependence among the rows.

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