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In linear algebra, the determinant is a special number that can be calculated from a square matrix. It is denoted as \( \text{det}(A) \) for a matrix \( A \). This number helps us determine if a matrix is invertible or non-invertible. A non-zero determinant indicates an invertible matrix. For example, in 2x2 matrices like matrix (a) from the exercise, calculating the determinant involves simple arithmetic across its diagonal elements:

• For \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is \( ad - bc \).

A non-zero result indicates the matrix is "full rank," meaning all rows are independent. In our exercise, matrix (a) had a determinant of 8, confirming its rank of 2. Determinants are powerful tools in understanding matrix behavior, especially for confirming matrix invertibility.

Row reduction is a systematic method for transforming a matrix into a simpler form, which helps determine key properties such as rank. This process involves elementary row operations:

• Swapping two rows.
• Multiplying a row by a non-zero scalar.
• Adding or subtracting a scalar multiple of one row from another row.

Through these operations, we simplify matrices while preserving their fundamental characteristics. In step 2 with matrix (b), we used row reduction to transform the matrix into a form where we could clearly identify the rank. By manipulating Rows 2 and 3 using Row 1, we clarified the distinct, independent rows, confirming the matrix rank as 3. Row reduction is a crucial technique in linear algebra, aiding both in calculation and comprehension of matrix properties.

Achieving row-echelon form is a key part of matrix analysis. A matrix is in row-echelon form when:

• Each leading entry (the first non-zero number from the left, in a non-zero row) is in a column to the right of the leading entry in the row above it.
• All entries below a leading entry are zeros.
• Any rows entirely composed of zeros are at the bottom of the matrix.

This form makes it straightforward to ascertain the rank of the matrix, as the number of non-zero rows corresponds directly to the rank. In matrix (c), row reduction led to a row-echelon form where three rows contained leading entries, signifying a rank of 3. Understanding the row-echelon form's structure is important for grasping matrix transformations and their implications.

An invertible matrix, also known as a non-singular or non-degenerate matrix, is one that possesses a multiplicative inverse. This means if you multiply the matrix by its inverse, you obtain the identity matrix. A key criterion for invertibility is that the determinant must be non-zero. In the exercise, we assessed the invertibility of matrix (a) by calculating its determinant. Since it was non-zero, the matrix is invertible. Invertibility is crucial as it indicates that the matrix describes a linear transformation with a unique reverse transformation. This property is particularly important when solving systems of linear equations or when attempting to find matrix solutions that revert to original data before transformation.