Which matrix has most of the elements (not all) as Zero?Select one:a.Identity Matrixb.Unit Matrixc.Sparse Matrixd.Zero MatrixClear my choice

Break Down the Problem

1. Identify the type of matrices provided in the options.
2. Understand the characteristics of each matrix type in relation to having elements as zero.

Relevant Concepts

1. Identity Matrix: A square matrix where all the diagonal elements are 1, and all other elements are 0.
2. Unit Matrix: Another term often used interchangeably with the identity matrix; typically refers to a matrix with 1s on the diagonal and 0s elsewhere.
3. Sparse Matrix: A matrix primarily composed of zero elements, which means that most of its elements are zero, but not necessarily all.
4. Zero Matrix: A matrix where all the elements are zero.

Analysis and Detail

1. Identity Matrix: For example, a $3 \times 3$ identity matrix looks like: $\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ Clearly, it has 3 out of 9 elements as non-zero.

2. Unit Matrix: Same as the identity matrix; it does not match the given criteria since it has non-zero elements.

3. Sparse Matrix:

   • Definition: A matrix with a high proportion of zero elements. For example: $\begin{pmatrix} 0 & 0 & 3 \\ 0 & 0 & 0 \\ 5 & 0 & 0 \end{pmatrix}$ In this case, there are many zeros compared to non-zero elements.

4. Zero Matrix: $\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$ All elements are zero; hence, it does not qualify as having "most" of the elements as zero.

Verify and Summarize

1. The identity and unit matrices have a few non-zero elements, while the zero matrix has none at all.
2. The sparse matrix is primarily composed of zeros but can have some non-zero elements, fitting the criteria of "most" elements being zero.

Final Answer

The correct answer is c. Sparse Matrix.