Question

If \(\operatorname{det} A=2\), det \(B=-1\), and \(\operatorname{det} C=\) 3, find: $$ \begin{array}{l} \text { a. } \operatorname{det}\left[\begin{array}{lll} A & X & Y \\ 0 & B & Z \\ 0 & 0 & C \end{array}\right] & \text { b. det }\left[\begin{array}{lll} A & 0 & 0 \\ X & B & 0 \\ Y & Z & C \end{array}\right] \\ \text { c. } & \text { det }\left[\begin{array}{lll} A & X & Y \\ 0 & B & 0 \\ 0 & Z & C \end{array}\right] & \text { d. det }\left[\begin{array}{ccc} A & X & 0 \\ 0 & B & 0 \\ Y & Z & C \end{array}\right] \end{array} $$

Short answer

a. -6, b. -6, c. -6, d. -6.

Step-by-step solution

Identify Block Matrix Structure for a

For part (a), the matrix is a block upper triangular matrix with main diagonal blocks \(A\), \(B\), and \(C\). The determinant of a block upper triangular matrix is the product of the determinants of the diagonal blocks.

Calculate Determinant for a

For part (a), the determinant is calculated as \( \operatorname{det}(A) \times \operatorname{det}(B) \times \operatorname{det}(C) \). Substitute the known determinants: \( 2 \times (-1) \times 3 = -6 \).

Identify Block Matrix Structure for b

For part (b), the matrix is a block lower triangular matrix with main diagonal blocks \(A\), \(B\), and \(C\). Similar to upper triangular, the determinant is the product of the determinats of the blocks on the main diagonal.

Calculate Determinant for b

For part (b), calculate as: \( \operatorname{det}(A) \times \operatorname{det}(B) \times \operatorname{det}(C) \). Using the given values: \( 2 \times (-1) \times 3 = -6 \).

Identify Block Matrix Structure for c

For part (c), the matrix is neither upper nor lower block triangular due to the block \(Z\) disrupting a direct product calculation. However, \(Z\) is in a position that won't affect the determinant because of the zeros above and below it.

Calculate Determinant for c

Use the same direct product calculation as before because the placement of \(Z\) doesn't affect the diagonal block structure: \( \operatorname{det}(A) \times \operatorname{det}(B) \times \operatorname{det}(C) = 2 \times (-1) \times 3 = -6 \).

Identify Block Matrix Structure for d

For part (d), the matrix retains a block upper triangular form where each diagonal component is unaffected by non-zero values off-diagonal.

Calculate Determinant for d

Use the block product theorem because the off-diagonal \(X\) doesn't affect the determinant: \( \operatorname{det}(A) \times \operatorname{det}(B) \times \operatorname{det}(C) = 2 \times (-1) \times 3 = -6 \).

Key concepts

Block Matrix

A block matrix is a large matrix divided into smaller matrices known as blocks or submatrices. These blocks can simplify complex matrix computations by breaking them into manageable pieces.
Block matrices are particularly useful in solving problems related to linear algebra and signal processing, where matrices may have components like zeros or submatrices that don't interact with one another.
Splitting matrices into blocks can help identify patterns, making computations easier. This approach allows for efficient calculation of determinants when the matrix can be divided into triangular blocks.

Upper Triangular Matrix

An upper triangular matrix is a square matrix where all entries below the main diagonal are zero. This structure makes calculations such as finding the determinant straightforward.
For an upper triangular matrix, the determinant is simply the product of the entries on the main diagonal. For example, in a 3x3 upper triangular matrix with diagonal elements a, b, c, the determinant is computed as: \( a \times b \times c \).
Because of this property, upper triangular matrices are especially useful in simplifying matrix equations, making them integral to computational functions like LU decomposition.

Lower Triangular Matrix

A lower triangular matrix is similar to its upper counterpart but flipped. In this case, all elements above the main diagonal are zeros. The determinant calculation is similarly straightforward to that of upper triangular matrices.
For a lower triangular matrix with diagonal elements d, e, f, the determinant is the product of these diagonal elements: \( d \times e \times f \).
This simplicity in computation makes lower triangular matrices valuable for solving systems of equations and matrix inverse operations, as it allows for elimination of variables with ease.

Matrix Determinant Properties

The determinant of a matrix is a special number that provides meaningful insights into various properties of the matrix. Here are some fundamental properties related to matrix determinants:

• For block triangular matrices, the determinant is the product of the determinants of the diagonal blocks.
• Switching two rows (or columns) of a matrix reverses its determinant's sign.
• If a matrix has a row or column of all zeros, its determinant is zero.
• Multiplying a matrix by a scalar multiplies its determinant by that same scalar, repeated according to the matrix's order.

These properties are essential in simplifying and solving matrix equations and understanding the matrix's geometric and algebraic implications.