# Matrix Theory: True or False Statements**Instructions:**Mark each statement T if the statement is always true or F if it’s ever false. All $ A $, $ B $, $ C $, and $ D $ are matrices, and $ I $ is the identity matrix. Do not assume anything beyond what is explicitly stated.**Statements:**(a) If $ A $ and $ B $ are $ m \times n $ matrices, then both products $ AB^\top $ and $ A^\top B $ are defined.(b) If $ C = D $, then $ BC = BD $.(c) If $ BC = BD $, then $ C = D $.(d) Every square matrix is the product of elementary matrices.(e) If $ AB = I $, then $ A $ is invertible.(f) For any $ A $, $ \text{det} (-A) = -\text{det} A $.(g) If $ A $ and $ B $ are square matrices with $ \text{det}(A) = 2 $ and $ \text{det}(B) = 3 $, then $ \text{det}(A + B) = 5 $.(h) If two rows of a square matrix $ A $ are the same, then $ \text{det} A = 0 $.(i) If $ \text{det} A = 5 $, then $ A $ is invertible and $ \text{det}(A^{-1}) = \frac{1}{5} $.(j) If $ A^2 = 0 $, then $ A $ is singular.### Detailed Explanations:* **Statement (a):** - **Analysis:** For the products $ AB^\top $ and $ A^\top B $ to be defined, the inner dimensions of the resulting matrices must match. Both matrices $ A $ and $ B $ are $ m \times n $, so $ B^\top $ (the transpose of $ B $) would be $ n \times m $. Thus, $ AB^\top $ would result in an $ m \times m $ matrix, and $ A^\top $ would be $ n \times m $ while $ B $ remains \( m \times n

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Mark each statement T if the statement is always true or F if it's ever false. All A, B, C, and D are matrices, and I is the identity matrix. Do not assume anything beyond what is explicitly stated. (a) If A and B are m × n matrices, then both products AB™ and A™B are defined. (b) If C = D, then BC = BD. %3D (c) If BC = BD, then C = D. (d) Every square matrix is the product of elementary matrices. (e) If AB = I, then A is invertible. (f) For any A, det(-A) = – det A. (g) If A and B are square matrices with det A = 2 and det B = 3, then det(A + B) = 5. (h) If two rows of a square matrix A are the same, then det A = 0. (i) If det A = 5, then A is invertible and det(A¯') =. (j) If A² = 0, then A is singular.

Mark each statement T if the statement is always true or F if it's ever false. All A, B, C, and D are matrices, and I is the identity matrix. Do not assume anything beyond what is explicitly stated. (a) If A and B are m × n matrices, then both products AB™ and A™B are defined. (b) If C = D, then BC = BD. %3D (c) If BC = BD, then C = D. (d) Every square matrix is the product of elementary matrices. (e) If AB = I, then A is invertible. (f) For any A, det(-A) = – det A. (g) If A and B are square matrices with det A = 2 and det B = 3, then det(A + B) = 5. (h) If two rows of a square matrix A are the same, then det A = 0. (i) If det A = 5, then A is invertible and det(A¯') =. (j) If A² = 0, then A is singular.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Concept explainers

Matrices

‘Matrices’ is the plural form of the word matrix, and it is basically a spreadsheet in the form of a box. In mathematics, various functions can be carried out with matrices. Generally, a matrix comes in the shape of a square or rectangle. The elements are distributed horizontally in the form of rows, and vertically in the form of columns.

Question

# Matrix Theory: True or False Statements

**Instructions:**
Mark each statement T if the statement is always true or F if it’s ever false. All $ A $, $ B $, $ C $, and $ D $ are matrices, and $ I $ is the identity matrix. Do not assume anything beyond what is explicitly stated.

**Statements:**

(a) If $ A $ and $ B $ are $ m \times n $ matrices, then both products $ AB^\top $ and $ A^\top B $ are defined.

(b) If $ C = D $, then $ BC = BD $.

(c) If $ BC = BD $, then $ C = D $.

(d) Every square matrix is the product of elementary matrices.

(e) If $ AB = I $, then $ A $ is invertible.

(f) For any $ A $, $ \text{det} (-A) = -\text{det} A $.

(g) If $ A $ and $ B $ are square matrices with $ \text{det}(A) = 2 $ and $ \text{det}(B) = 3 $, then $ \text{det}(A + B) = 5 $.

(h) If two rows of a square matrix $ A $ are the same, then $ \text{det} A = 0 $.

(i) If $ \text{det} A = 5 $, then $ A $ is invertible and $ \text{det}(A^{-1}) = \frac{1}{5} $.

(j) If $ A^2 = 0 $, then $ A $ is singular.

### Detailed Explanations:

* **Statement (a):**
- **Analysis:** For the products $ AB^\top $ and $ A^\top B $ to be defined, the inner dimensions of the resulting matrices must match. Both matrices $ A $ and $ B $ are $ m \times n $, so $ B^\top $ (the transpose of $ B $) would be $ n \times m $. Thus, $ AB^\top $ would result in an $ m \times m $ matrix, and $ A^\top $ would be $ n \times m $ while $ B $ remains \( m \times n

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